control strategies and parameter compensation for permanent

Control Strategies and ParameterCompensation for Permanent Magnet

Synchronous Motor Drives

Ramin Monajemy

Dissertation Submitted to the Faculty of the Virginia Polytechnic Institute andState University in Partial Fulfillment of the Requirements for the degree of

Doctor of PhilosophyIn

Electrical Engineering

Krishnan Ramu, ChairmanHugh VanLandinghamWilliam T. BaumannCharles E. NunnallyLance A. Matheson

October 12, 2000Blacksburg, Virginia

Key Words: Motor Drives, Permanent Magnet Synchronous Motor, ControlStrategies, Operational Boundary, Power Losses

Copyright 2000, Ramin Monajemy

Control Strategies and Parameter Compensation forPermanent Magnet Synchronous Motor Drives

Ramin Monajemy

(Abstract)

Variable speed motor drives are being rapidly deployed for a vast range of applications in order

to increase efficiency and to allow for a higher level of control over the system. One of the

important areas within the field of variable speed motor drives is the system’s operational

boundary. Presently, the operational boundaries of variable speed motor drives are set based on

the operational boundaries of single speed motors, i.e. by limiting current and power to rated

values. This results in under-utilization of the system, and places the motor at risk of excessive

power losses. The constant power loss (CPL) concept is introduced in this dissertation as the

correct basis for setting and analyzing the operational boundary of variable speed motor drives.

The control and dynamics of the permanent magnet synchronous motor (PMSM) drive operating

with CPL are proposed and analyzed. An innovative implementation scheme of the proposed

method is developed. It is shown that application of the CPL control system to existing systems

results in faster dynamics and higher utilization of the system. The performance of a motor drive

with different control strategies is analyzed and compared based on the CPL concept. Such

knowledge allows for choosing the control strategy that optimizes a motor drive for a particular

application. Derivations for maximum speed, maximum current requirements, maximum torque

and other performance indices, are presented based on the CPL concept. High performance

drives require linearity in torque control for the full range of operating speed. An analysis of

concurrent flux weakening and linear torque control for PMSM is presented, and implementation

strategies are developed for this purpose. Implementation strategies that compensate for the

variation of machine parameters are also introduced. A new normalization technique is

introduced that significantly simplifies the analysis and simulation of a PMSM drive's

performance. The concepts presented in this dissertation can be applied to all other types of

machines used in high performance applications. Experimental work in support of the key

claims of this dissertation is provided.

iii

Dedication

I dedicate this dissertation to my father, mother and sister who believed in me all through these

years, and provided outstanding moral support.

iv

Acknowledgements

I would like to thank my advisor, Professor Krishnan Ramu, for providing guidance and sharing

his vision and experiences throughout the development of this dissertation. Many thanks also

goes to my committee members, Professor VanLandingham, Professor Baumann, Professor

Nunnally and Professor Matheson, for monitoring this effort and for providing their valuable

opinions.

I would like to thank the members of the MCSRG group for the many discussions that we had

and for sharing their opinion on different subjects. I would like to acknowledge and thank

former Ph.D. student, Dr. Nishith Tripathi, for assisting me in preparing and training the neural

networks that are used in this dissertation.

v

Table of Contents

List of Figures x

Symbols and Abbreviations xiii

1 Introduction 1

1.1 Foreword 1

1.2 Areas of Interest and Research Outline 2

1.3 Drawbacks of Existing Control Techniques 4

1.3.1 Under-utilization in the Lower than Base Speed Range 4

1.3.2 Excessive Power Losses in the Flux Weakening Range 5

1.3.3 Torque Non-linearity Caused by Ignoring Core Losses 5

1.3.4 Torque Non-linearity Caused by Improper Implementation 5

1.3.5 Confusion in the Definition of Base Speed 6

1.4 Advantages of the Techniques Presented in this dissertation 6

1.5 Potential Impact on the Research Community and Industry 7

1.6 Dissertation Structure 7

1.7 Summary of Contributions and List of Co-authored Papers 8

1.8 Assumptions 9

1.9 Conclusions 10

2 State of the Art 11

2.1 Introduction 11

2.2 The Electric Motor: A Historical Perspective 11

2.3 Trends Forcing the Transition to Variable Speed Drives 15

2.4 Why Concentrate this Study on PMSM? 17

2.5 Literature Review 17

2.5.1 Power Losses and Efficiency 18

2.5.2 Operational Boundaries of Motor Drives 19

2.5.3 Control Strategies for Operation Below Base Speed 19

2.5.4 Control Strategies for Operation Above Base Speed 21

vi

2.5.5 Parameter Sensitivity and Parameter Insensitive Control Strategies 22

2.6 Conclusions 23

3 Control and Dynamics of Constant Power Loss Based Operation of PMSM 24

3.1 Introduction 24

3.2 PMSM Model with Losses 26

3.2.1 Electrical Equations of PMSM Including Core Losses 27

3.2.2 Total Power Loss Equation for PMSM 28

3.3 Constant Power Loss Control Scheme and Comparison 28

3.3.1 The Lower than Base Speed Operating Range 31

3.3.2 The Flux Weakening Operating Range 31

3.4 Secondary Issues of the CPL Controller 32

3.4.1 Higher Current Requirement at Lower than Base Speed 32

3.4.2 Parameter Dependency 32

3.5 Implementation Scheme for the CPL Controller 33

3.6 Experimental Correlation 34

3.7 The Drive Control with Power Loss Command 35

3.7.1 Cycling the Drive 35

3.7.2 Commanding Short Term Maximum Torque 36

3.8 Conclusions 38

4 The CPL Controller for Applications with Cyclic Loads 39

4.1 Introduction 39

4.2 The Power Loss Command in Different Application Categories 40

4.2.1 Derivation of a Practical Power Loss Estimator 40

4.2.2 Power Loss Command and Maximum Torque in

Different Application Categories 41

4.3 Comparison of Maximum Torque in Different Load Categories 51

4.3.1 Load Categories A to D 51

4.3.2 Load Category E 53

4.4 Conclusions 53

vii

5 Performance Evaluation and Comparison of Control Strategies 55

5.1 Introduction 55

5.2 Performance Criteria 57

5.3 Control Strategies: Lower than Base Speed Operating Range 59

5.3.1 Maximum Efficiency Control Strategy 61

5.3.2 Zero D-Axis Current Control Strategy 62

5.3.3 Maximum Torque per Unit Current Control Strategy 63

5.3.4 Unity Power Factor Control Strategy 63

5.3.5 Constant Mutual Flux Linkages Control Strategy 64

5.4 Control Strategies: Higher than Based Speed Operating Range 65

5.4.1 Constant Back EMF Control Strategy 66

5.4.2 Six Step Voltage Control Strategy 69

5.5 Comparison of Control Strategies Based on the CPL Concept 74

5.5.1 Lower than Base Speed Operating Range 74

5.5.2 Higher than Base Speed Operating Range 80

5.6 Performance Comparison inside the Operational Envelope 84

5.6.1 Current vs. Torque 85

5.6.2 Air Gap Flux Linkages vs. Torque 85

5.6.3 Power Factor vs. Torque 85

5.6.4 D-Axis Current vs. Torque 87

5.6.5 Torque Range 87

5.7 Direct Steady State Evaluation in SSV Mode 87

5.8 Simulation and Experimental Verification 91

5.9 Conclusions 93

6 Analysis and Implementation of Concurrent Mutual Flux Weakening and Torque

Control for PMSM 95

6.1 Introduction 95

6.2 Mutual Flux Linkages Weakening Strategy and Control 96

6.2.1 The Mutual Flux Linkages Weakening Strategy 96

6.2.2 Mutual Flux Linkages and Torque Control 96

viii

6.3 Maximum Torque for a Given Mutual Flux Linkages 97

6.4 Controller Block Diagram 98

6.5 Implementation Strategies 100

6.5.1 Lookup Table Approach 100

6.5.2 Two-Dimensional Polynomials Approach 103

6.6 Comparison of the Two Approaches 105

6.7 Conclusions 106

7 Concurrent Mutual Flux Weakening and Torque Control in the Presence of Parameter

Variations 107

7.1 Introduction 107

7.2 The Variation of Parameters in PMSM 108

7.3 The Impact of Parameter Variations 109

7.4 Parameter Compensation Scheme 111

7.4.1 Lookup Table Approach 112

7.4.2 Artificial Neural Network Approach 113

7.5 Comparison of the Two Approaches 114

7.6 Dynamic Simulation 115

7.7 Estimation of Machine Parameters 117

7.8 Conclusions 118

8 Applications of a New Normalization Technique 119

8.1 Introduction 119

8.2 The New Normalization Technique 120

8.3 Comparison to other Normalization Techniques 122

8.4 Derivation of Maximum Torque vs. Speed Envelopes 123

8.4.1 The Maximum Efficiency Control Strategy 123

8.4.2 The Maximum Torque per Unit Current Control Strategy 125

8.4.3 The Zero D-Axis Current Control Strategy 125

8.4.4 The Unity Power Factor Control Strategy 126

8.4.5 The Constant Mutual Flux Linkages Control Strategy 126

ix

8.4.6 The Constant Back EMF Control Strategy 127

8.5 Derivation of Maximum Current Requirement with CPL 128

8.6 Maximum Possible Torque as a Function of Flux Linkages 129

8.7 Generalized Performance Characteristics 131

8.8 Conclusions 135

9 Conclusions and Recommendations for Future Work 136

9.1 Conclusions 136

9.2 Recommendations for Future Work 138

Appendix I Prototype PMSM Drive 139

Appendix II Measurement of PMSM Parameters 140

References 143

Vita 150

x

List of Figures

Figure 3.1. q and d axis steady-state model in rotor reference frame includingstator and core loss resistances. 27

Figure 3.2. Normalized maximum torque, power loss, air gap power, voltageand phase current vs. speed for the CPLC (solid lines) and for thescheme with current and power limited to rated values(dashed lines). 30

Figure 3.3. Implementation scheme for the constant power loss controller. 33

Figure 3.4. Current, torque, estimated torque, total power loss, copper andcore losses along the CPLC boundary with power loss referenceof 30 W. 36

Figure 3.5. Dynamic response of the system for a step torque command of1.5 N.m. 37

Figure 3.6. Simulated speed, mω (RPM), and phase current command,*aI (A), in response to 1.5 N.m. step torque command. 37

Figure 4.1. Normalized torque, eT , and speed, rω , profiles for continuous

operation. 42

Figure 4.2. Normalized torque and speed profiles for on-off operation withsmall transition times. 44

Figure 4.3. Normalized torque and speed profiles for speed reversal application. 46

Figure 4.4. Normalized torque and speed profiles for operation between rmω± . 47

Figure 4.5. Normalized torque and speed profiles for operation with significanttransition times. 49

Figure 4.6. Maximum torque vs. maximum power loss for load categories A, B, C and D forduty cycles equal to 1, 0.75, 0.5, 0.33. 52

Figure 5.1. Normalized motor variables in steady state for a six step voltage inputvs. rotor angle in electrical degrees at 1635 RPM and dc bus voltageof 65 Volts and voltage phasor angle of 116 degrees. 73

xi

Figure 5.2. Maximum torque, current, power and net loss vs. speed at ratedpower loss for lower than base speed control strategies, and

1.1Em = p.u. for the flux weakening range. 76

Figure 5.3. Torque per current, back emf, power factor and net loss vs. speedat rated power loss for lower than base speed control strategies,and 1.1Em = p.u. for the flux weakening region. 77

Figure 5.4. Maximum torque, peak current, net power loss, torque ripple asa percentage of average torque, and peak to peak torque ripplefor SSV (solid lines) and CBE (dashed lines, mE =0.8 p.u.)control strategies. 82

Figure 5.5. Current, air gap flux linkages, power factor and d-axis currentvs. torque for different control strategies. ( 85.2L/L dq = ) 86

Figure 5.6. Estimated (dashed lines) and measured (solid lines) maximumtorque vs. speed trajectory for the SSV and CBE controlstrategies with maximum power loss of 30 W and .V65Vdc = 92

Figure 5.7. Measured phase voltage and current for the SSV control strategyat 1635 RPM, 116=α degrees and =dcV 65 V. 92

Figure 5.8. Simulated phase voltage and current for the SSV control strategyat 1635 RPM, 116=α degrees and =dcV 65 V. 93

Figure 6.1. Maximum permissible torque command as a function of mutualflux linkages command. 98

Figure 6.2. Controller block diagram for the flux weakening range. 99

Figure 6.3. Controller block diagram for the lower than base speed operatingrange. 99

Figure 6.4. *qni as a function of normalized mutual flux linkages and torque

commands. 102

Figure 6.5. *dni as a function of normalized mutual flux linkages and torque

commands. 102

Figure 6.6. Normalized speed, torque, mutual flux linkages, and phasevoltage for a ± 3 p.u. step speed command. 104

xii

Figure 6.7. Commanded and achieved mutual flux linkages vs. commandedtorque for the polynomial implementation approach. 105

Figure 7.1. The mutual flux linkages and torque control system. 109

Figure 7.2. Normalized torque error when p.u.8.0af =λ 111

Figure 7.3. Mutual flux linkages error when p.u.8.0af =λ 111

Figure 7.4. The mutual flux linkages and torque control with compensationfor a varying parameter. 112

Figure 7.5. The ANN based offset generator structure. 114

Figure 7.6. Commanded and actual values for speed, torque, mutual fluxlinkages, q and d-axis currents and phase voltage for a ± 3 p.u. speedcommand in the presence of 20 percent reduction in rotor flux linkages. 116

Figure 7.7. Commanded and actual values for mutual flux linkages and torque for a± 3 p.u. speed command in the presence of a 20 percentreduction in q-axis inductance. 117

Figure 8.1. Generalized application characteristics for PMSM with 3,2,1=ρ for the 132maximum torque per unit current control strategy.

Figure 8.2. Generalized application characteristics for PMSM with 3,2,1=ρ for the 134zero d-axis current control strategy.

* Any of the variables shown here can be used with a subscript n to denote normalization.

Symbols and Abbreviations

α angle of stator voltage phasor with respect to rotor's d-axisd duty cycleE back emf

mE maximum desired back emf

Er

back emf phasorη efficiency

I torque generating portion of stator current magnitude (steady state)

Ir

current phasor

bI base value for current

cI core loss portion of stator current magnitude (steady state)

dq I,I q and d axis torque generating currents, respectively (steady state)

dcqc I,I q and d axis core loss currents, respectively (steady state)

dmqm I,I maximum q and d axis currents, respectively

dsqs I,I q and d axis stator currents, respectively (steady state)

sI stator current magnitude (steady state)

smI maximum stator current magnitude

srI rated current

ai phase a current*ai phase a current command

c,b,ai phase a, b, c currents

dsqs i,i q and d axis stator currents, respectively

*ds

*qs i,i q and d axis stator current commands, respectively

*ds

*qs i,i nominal q and d axis stator current commands, respectively

dq i,i q and d axis torque generating currents, respectively

si stator current magnitude*d

*q i,i ∆∆ q and d axis current command offsets, respectively

tK motor torque constant

dq L,L q and d-axis inductances, respectively

bL base value for inductance

xiv

mλ mutual (or air gap) flux linkages

bλ base value for flux linkages*mλ mutual (or air gap) flux linkages command

afλ rotor flux linkages

afeλ rotor flux linkages estimation

afλ nominal value of rotor flux linkages

mλ∆ mutual flux linkages error

rω electrical speed*rω commanded speed

rrω rated speed

bω base value for speed

mω measured speed

rmω maximum speed

lmP maximum permissible power loss

ldlq P,P q and d axis power losses, respectively

lcP total core losses*lP power loss command

lP power losses

lP average power losses

lfP filtered power losses estimation

aP air gap power

rP rated power

bP base value for power

P number of poles

lrP total copper and core losses at rated speed and torque

sR , cR stator and core loss resistors, respectively

ρ ratio of q and d axis inductances (saliency ratio)T cycle period

offT off time (in one period)

onT on time (in one period)

1pT∆ rise time

2pT∆ fall time

mT∆ time during which constant torque is applied

zT∆ time during which no torque is applied

eT air gap torque

xv

*eT commanded torque*eT

~commanded torque after being limited

ebT base value for torque

emT maximum air gap torque

epT peak air gap torque

limT torque limit

eT∆ torque error

mθ measured rotor position

rθ rotor electrical position

bV base value for voltage

dcV dc bus voltage

mV fundamental component of stator voltage magnitude

dsqs V,V q and d axis stator voltages, respectively (steady state)

sV stator voltage magnitude (steady state)

smV maximum stator voltage magnitude

av phase a voltage

dsqs v,v q and d axis stator voltages, respectively

lW energy losses in one period

BDCM brushless dc motorCBE constant back emf (control strategy)CMFL constant mutual flux linkages (control strategy)CPL constant power lossCPLC constant power loss controllerDAC digital to analog converterMTPC maximum torque per unit current (control strategy)PI proportional integrator (controller)PMSM permanent magnet synchronous motorpf power factorSSV six-step voltage (control strategy)UPF unity power factor (control strategy)ZDAC zero d-axis current (control strategy)

1

CHAPTER 1

Introduction

1.1 FOREWORD

For the better part of the 20th century most motion control systems were designed to operate at

a fixed speed. Many existing systems still operate based on a speed determined by the frequency

of the power grid. However, the most efficient operating speed for many applications, such as

fans, blowers and centrifugal pumps, is different from that enforced by the grid frequency. Also,

many high performance applications, such as robots, machine tools and the hybrid vehicle,

require variable speed operation to begin with. As a result, a major transition from single speed

systems to variable speed systems is in progress. The transition from single speed drives to

variable speed drives has been in effect since the 1970s when movements towards conservation

of energy and protection of the environment were initiated. About seventy percent of all

electrical energy is converted into mechanical energy by motors in the industrialized world. This

may be the most important factor behind today’s high demand for more efficient motion control

systems. A large body of research is available on variable speed drives due to the significant

industrial and commercial interest in such systems. However, some areas of importance merit

further investigation. One such area is the operational boundary of motor drives. The

operational boundaries of variable speed motor drives are being incorrectly set based on the

operational boundaries of single speed motor drives, i.e. by limiting current and power to rated

values. The operational boundary of any motor drive must be set based on the maximum

possible power loss vs. speed profile for the given motor. Also, all control strategies for a

machine must be analyzed and compared based on such an operational boundary.

The areas of interest in this dissertation are discussed in section 1.2. The drawbacks of existing

control techniques for high performance control of PMSM are discussed in section 1.3. The

advantages of the techniques developed in this dissertation are discussed in section 1.4. The

potential impact of this research effort on the research community and on the industry is

Chapter 1 Introduction 2

discussed in section 1.5. The structure of this dissertation is presented in section 1.6. The

summary of key contributions of this dissertation are given in section 1.7. The assumptions are

discussed in section 1.8. The conclusions are summarized in section 1.9.

1.2 AREAS OF INTEREST AND RESEARCH OUTLINE

The emphasis of this dissertation is on high performance control strategies for variable speed

motor drives. High performance control strategies are capable of providing accurate control over

torque or speed to within a small percentage error. A high performance control strategy can also

optimize one or more performance indices such as torque, efficiency and power factor. The

following areas are of particular interest within the scope of high performance motor drives:

• Operational boundary of motor drives

• Implementation strategies that automatically limit the operational boundary for the full range

of operational speed based on a desired power loss profile

• Performance of different torque control strategies based on operation under constant power

loss

• Implementation strategies for concurrent flux weakening and torque control

• Parameter insensitive control strategies for concurrent flux weakening and torque control

• Innovative normalization techniques that simplify the analysis and simulation of motor drive

performance

The operational boundary of a machine is generally defined by the rated current and power of

the machine. This operational boundary is only valid at rated speed. However, the same

operational boundary has been wrongly carried over to variable speed motor drives. The true

operational boundary of a machine depends on the maximum permissible power loss profile for

the machine. Copper and core losses are the most fundamental sources of power losses in a

machine. Core losses are more significant than copper losses at higher speeds. A

comprehensive study of the operational boundaries of motor drives is performed in this

dissertation. Both copper and core losses are taken into account. The constant power loss (CPL)

concept is introduced here as a basis for defining the operational boundary of a motor drive. The

control and dynamics of the permanent magnet synchronous motor (PMSM) drive operating


under CPL is proposed and evaluated in this dissertation. An innovative implementation scheme

for the proposed method is developed. It is shown that application of the CPL control system

results in faster dynamics and higher utilization of existing motor drives. It is also shown that

the traditional method of defining the operational boundary results in under-utilization of the

machine and can put the system at risk of excessive power losses. Another area where further

research is warranted is the analysis and comparison of the wide variety of control strategies

available for high performance motor drives. The main control strategies for the lower than base

speed operating range for PMSM are the maximum efficiency, maximum torque per unit current,

zero d-axis current, unity power factor and constant mutual flux linkages. The main control

strategies for the higher than base speed operating range are constant back emf and six step

voltage. A comprehensive analysis and comparison of these control strategies has not been made

to this date. Availability of such analysis and comparison is the key to choosing a control

strategy that optimizes the operation of a particular motion control system. All control strategies

are analyzed and compared in this dissertation based on the constant power loss concept that

defines the operational boundary in each case. The comparison is based on the performance of

the system along the constant power loss operational boundary, and also based on

implementation requirements.

The proposed CPL controller is built around a wide speed range linear torque controller.

Implementation techniques for linear torque control strategies for the lower than base speed

range have been proposed and studied in the literature. Implementation techniques for linear

torque control in the higher than base speed operating range are discussed in this dissertation.

The proposed implementation techniques are based on lookup tables or equations. Torque

control in the higher than base speed operating range implicitly involves the control of mutual

flux linkages. The limitation of such a system is analyzed. The new implementation techniques

lend themselves to the implementation of parameter insensitive torque controllers. Parameter

insensitive controllers for concurrent flux weakening and torque control for PMSM drives are

proposed. Implementation schemes for these controllers are based on lookup tables or artificial

neural networks.

All the analysis and simulations presented in this dissertation have been performed using an

innovative normalization technique. This technique, along with a number of its applications, is


discussed in the Chapter 8. However, the equations and derivations in the earlier chapters are

presented in non-normalized form so that the analysis and results can be easily comprehended by

researchers as well as industrial engineer.

This study lays the foundation for the analysis, development and implementation of truly

optimized motor drives for wide speed range motion control systems based on PMSM. Similar

techniques can be applied to all types of motor drives. In the next section the drawbacks of

existing control techniques are summarized and discussed.

1.3 DRAWBACKS OF EXISTING CONTROL TECHNIQUES

The drawbacks of existing control techniques and strategies for PMSM drives, as far as

operational boundary and torque control are concerned, are briefly discussed in this section. The

control techniques developed in this dissertation address these drawbacks. The list of these

drawbacks is given below:

• Under-utilization of the motor in the lower than base speed range of operation

• Possibility of excessive power losses at higher than rated speeds

• Torque non-linearity as speed increases

• Torque non-linearity as a result of improper implementation

• Confusion in the definition of base speed

These drawbacks are discussed below.

1.3.1 Under-utilization in the Lower than Base Speed Range

As discussed earlier, limiting the current of a machine to the rated value for lower than rated

speeds results in under-utilization of the machine. Rated current defines the correct current limit

at rated speed only. This magnitude of current results in a specific magnitude of copper losses.

The sum of the copper and core losses at rated speed is equal to the maximum permissible loss at

rated speed. However, as speed decreases core loss decrease significantly. Therefore, a higher

magnitude of copper losses can be tolerated at lower than rated speed. This means that a higher

than rated magnitude of current can be tolerated at lower than rated speeds.


1.3.2 Excessive Power Losses in the Flux Weakening Range

It is common to limit a machine's power to its rated value in the flux weakening range.

However, limiting the power of the machine to its rated value does not guarantee that the power

losses remain limited. This is due to the fact that core losses increase significantly at higher than

rated speeds. It is shown in Chapter 3 that the net loss can exceed the maximum permissible loss

at a certain speed above rated speed despite the fact that power is maintained at rated level.

1.3.3 Torque Non-linearity Caused by Ignoring Core Losses

Another drawback of most existing control strategies is that they rely on an electrical model of

the machine that does not account for core losses. As speed increases a larger percentage of the

input current is utilized in the generation of core losses. Obviously, this fraction of the input

current does not contribute to the generation of torque. Therefore, the assumption that the entire

input current generates torque at all speeds results in torque non-linearity as speed approaches

the rated speed and beyond.

It is important to realize that core losses constitute a significant portion of the net losses at

speeds close to, and higher, than rated speed. Therefore, any model-based high performance

control strategy must use a model that takes core losses into account. This is specially true if the

system requires linear control of torque at higher than base speeds.

1.3.4 Torque Non-linearity Caused by Improper Implementation

Many implementation schemes presented in the literature assume that the magnitude of current

and torque are proportional for PMSM. In these cases, linear control over current is

implemented in the drive, assuming that linear control over torque follows automatically.

However, this assumption is only valid for some types of PMSM such as the surface mount

PMSM. But, strictly speaking, the relationship between torque and current is not linear for

PMSM. This is specially true for inset and interior PMSM.


1.3.5 Confusion in the Definition of Base Speed

The base speed of a machine is defined here as the speed after which generation of maximum

torque requires application of maximum voltage. It is common among traditional high

performance control techniques to use the rated speed as the base speed. Using the rated speed

as the transition point to the flux weakening range is proper only for the surface mount PMSM

running in a nominal environment at rated voltage. It is shown in Chapter 5 that the base speed

is a function of maximum permissible power loss, bus voltage, and the choice of control strategy.

Therefore, simply defining the base speed as rated speed results in improper operation of the

motor drive.

1.4 ADVANTAGES OF THE TECHNIQUES PRESENTED IN THIS DISSERTATION

The application of the analysis and schemes provided in this dissertation result in motor drive

systems with the following advantages:

• Faster dynamics and increased productivity

• Cost savings by maximizing machine utilization

• Optimization of the system based on application’s requirements

• Protection of the machine from excessive power losses

• Adaptability of the motor drive to different environment without changing existing design

• Simple and effective linear torque controllers for wide speed range motor drives

The techniques developed in this dissertation result in maximum utilization of a motor drive

under all operational conditions. This means that more torque and power can be derived from

existing motor drives, resulting in faster dynamic response and increased productivity. On the

contrary, if an application does not require more torque then a motor of smaller size and lower

price can be used. In addition, application of the schemes described in this dissertation result in

operation of the machine within the thermal boundaries imposed by the motor structure and

environmental conditions. Generally, the motor drive becomes more adaptable to different

operating conditions.


1.5 POTENTIAL IMPACT ON THE RESEARCH COMMUNITY AND INDUSTRY

The analytical procedures and implementation techniques, presented in this dissertation, in

conjunction with existing knowledge base, are expected to have the following impacts on the

research and industrial communities:

• Operational boundary of all types of motor drives will be defined and studied based on the

constant power loss (CPL) concept.

• Drives will be designed to fully utilize an existing motor’s capacity to make the system

adaptable to different environment by utilizing CPL controllers.

• Similar procedures will be applied to the permanent magnet brushless dc, synchronous

reluctance, switched reluctance and other motors. The permanent magnet brushless dc motor

is an immediate beneficiary since it is widely used in low power high performance

applications.

The content of this dissertation and associated papers can be used as reference on the subjects

discussed above.

1.6 DISSERTATION STRUCTURE

Chapter 2 provides a historical account of motor drives as far as this study is concerned. A

comprehensive literature review in related areas is also given. The control and dynamics of the

PMSM drive operating with constant power loss is proposed and analyzed in chapter 3. A

comparison to traditional operational methods is given. The CPL controller for applications with

cyclic loads is discussed in Chapter 4. The performance of different control strategies for wide

speed range operation of PMSM is analyzed and compared under the CPL concept in Chapter 5.

Implementation strategies for the concurrent flux weakening and torque control of PMSM are

introduced and analyzed in chapter 6. Implementation strategies for the concurrent flux

weakening and torque control of PMSM in the presence of parameter variations are introduced in

Chapter 7. A new normalization technique, along with its applications, is presented in Chapter 8.

The conclusions, summary of contributions, and recommendations for future work are provided

in Chapter 9. Appendix I describes the parameters of the PMSM drive prototype that was used


to verify key results of this study. Appendix II describes the procedures used in measuring the

various parameters of the PMSM drive described in Appendix I.

The new normalization technique described in Chapter 8 is used in the analysis of system

performance, and in preparing the code for the simulations, presented in this dissertation. This

normalization technique simplifies the analysis and simulation of system performance.

However, unless otherwise noted, all equations and derivations in this dissertation are presented

as non-normalized statements so that both industrial engineers and researchers can benefit

equally. All variables in plots are normalized using rated values for the respective variables to

enhance the overall presentation and also to simplify the process of generalization of the results.

All simulation codes have been prepared using MATLAB.

1.7 SUMMARY OF CONTRIBUTIONS AND LIST OF CO-AUTHORED PAPERS

The key contributions of this dissertation are outlined below:

• Constant power loss based control strategy to obtain the maximum torque vs. speed envelope

• Its comparison to schemes with current and power limits

• An implementation scheme for the proposed CPL controller and its flexibility for

incorporation into existing drives that may have various control scheme realizations in its

torque and flux controllers

• Derivation of maximum torque and power loss command as a function of maximum

permissible power loss, load duty cycle and speed for applications with cyclic loads

• Comparison of maximum possible torque as a function of maximum possible power loss for

different applications with cyclic loads

• Performance evaluation and comparison of control strategies for the full range of operating

speed based on operation with constant power loss and implementation requirements

• Analytical derivation of maximum speed, maximum current requirements of the drive and

maximum current in the flux weakening range based on the constant power loss criteria

• Analytical derivation of the maximum possible torque as a function of mutual flux linkages

• Implementation schemes for the concurrent flux weakening and torque control for PMSM,

and parameter insensitive controllers based on this concept


• Introduction of a new normalization technique that simplifies the analysis and simulation of

PMSM drive operation

• Derivation of the maximum torque vs. speed envelopes for the ME, MTPC, ZDAC, CMFL,

UPF, CBE control strategies operating with constant power loss

• Introduction of generalized application characteristics for PMSM drives using the new

normalization technique

• Experimental verification of key results

List of author’s publications relevant to this dissertation:

•••• R. Monajemy and R. Krishnan, “Control and Dynamics of Constant Power Loss Based

Operation of Permanent Magnet Synchronous Motor,” Conference record, IEEE Industrial

Electronics Conference (IECON), Nov. 29-Dec. 3, 1999, pp. 1452-1457.

•••• R. Monajemy and R. Krishnan, “Performance comparison of six-step voltage and constant

back emf control strategies for PMSM,” Conference record, IEEE Industry Applications

Society Annual Meeting, Oct. 1999, pp. 165-172.

•••• R. Monajemy and R. Krishnan, “Concurrent mutual flux and torque control for the

permanent magnet synchronous motor,” Conference record, IEEE Industry Applications

Society Annual Meeting, Oct. 1995, pp. 238-245.

•••• R. Krishnan, R. Monajemy and N. Tripathi, “Neural Control of High Performance Drives:

An Application to the PM Synchronous Motor Drive”, Invited Paper, Conference records,

IEEE Industrial Electronics Conference (IECON), November, 1995, pp. 38-43.

1.8 ASSUMPTIONS

The following assumptions are made so that the fundamentals of this study can be presented

with better clarity.

• All motor-drive parameters are assumed to be constant unless otherwise noted

• Leakage inductances are zero

• Windage and friction are negligible


• Net sustainable loss for a machine is assumed to be constant for the full operating range for

simulation and analytical purposes

• A high band-width current controller is utilized in the drive system resulting in negligible

stator current error

• The rated current is defined as the current that generates rated torque under the zero d-axis

current control strategy

• Base speed is the speed after which flux weakening becomes necessary along the CPL

boundary

1.9 CONCLUSIONS

The operational boundary of any motor drive needs to be set and analyzed based on the

maximum permissible power loss vs. speed profile for the given machine. The traditional

method of defining the operational boundary by limiting torque and power to rated values results

in under-utilization of the system and may subject the motor to excessive power losses. The

analysis of the operational boundary of a motor drive based on the concept of constant power

loss is the main topic of this dissertation. An implementation strategy for enforcing the CPL

operational boundary for a motor drive is presented in this dissertation. The performance of all

advanced control strategies for permanent magnet synchronous motors are analyzed and

compared based on operation of the motor under the CPL concept. Implementation strategies for

concurrent flux weakening and torque control for PMSM are presented. Implementation

strategies for concurrent flux weakening and torque control in the presence of parameter

variations are also presented. All simulation and analysis are performed using an innovative

normalization technique that significantly simplifies the research effort. Experimental

verification of the key theoretical claims of this dissertation are provided using a prototype

PMSM drive described in Appendix I.

11

CHAPTER 2

State of the Art

2.1 INTRODUCTION

In this chapter the rationale behind the research presented in this dissertation is provided.

Permanent magnet synchronous motors and drives are put into perspective from a historical point

of view. The ongoing transition from single-speed to variable-speed motor drives is discussed.

The trends behind this transition are studied. This transition has resulted in the need for

revisiting the area of operational boundaries for motor drives. The state of the art in the areas of

computation of power losses, operational boundaries for variable speed motor drives, control

strategies for high performance variable speed motor drives, and parameter insensitive control

strategies for PMSM are given in this chapter. This chapter justifies the need for the research

effort presented in this dissertation.

Section 2.2 provides a brief history of motors with emphasis on the evolution of PMSM drives.

The period from early 1800s to the 1970s is covered. The trends of the last three decades that are

behind the transition from single speed to variable speed drives are discussed in section 2.3. The

rationale behind choosing the PMSM as the immediate beneficiary of this research effort is

discussed in section 2.4. A literature review on the relevant topics of this dissertation is provided

in section 2.5. The conclusions of this chapter are given in section 2.6.

2.2 THE ELECTRIC MOTOR: A HISTORICAL PERSPECTIVE

In this section motors and motor drives are put into perspective from a historical point of view.

The evolution of the PMSM is traced back in time, and the developments that allow high

performance control over synchronous motors are reviewed.

English physicist and chemist Michael Faraday constructed a primitive model of the electric

motor in 1821. This was a dc motor [1]. By the early 1870s the Belgian-born electrical engineer

Zénobe-Théophile Gramme had developed the first commercially viable dc motor. The dc motor

Chapter 2 State of the Art 12

was in wide spread use in street railways, mining and industrial applications by the year 1900. A

book titled “The Electric Motor and its Applications” [2], published in 1887, is an indication of

the widespread interest in the electric motor in those years. In that book the general theoretical

background necessary for understanding dc motor operation is given. Various types of dc

motors, available at that time, along with their numerous applications and control ideas, with

applications in public transportation, are discussed. The concept of controlled delivery of energy

from a dc power source to a motor in order to meet the specific demands of an application is

discussed. The same process is used today in the majority of high performance control systems.

The following paragraph is a quote from [2], Chapter VII, page 99:

“The Use of Storage Batteries with Electric Motors for Street Railways”

In the present chapter we take up a method which, although now looked upon with distrust by

many, may yet prove to be one of the most feasible means for propulsion of railway cars. We

refer to the employment of accumulators, the stored energy of which, conveyed to the motor in

the form of current, sets it in motion, and with it the car. While this mode of propulsion was until

lately in the experimental state, the progress made has been such that a satisfactory solution of

the problem appears to have been reached; indeed the immediate future will see cars propelled

by the energy derived from accumulators, with success, judged from the standpoint both of

convenience and economy. (T.C. Martin and J. Wetzler, 1887)

However, mass production of combustion engines in the early 1900s, and the abundance of

fossil fuels, delayed the use of accumulators in transportation applications until the present time.

Soon, the disadvantages of dc motors, such as excessive wear in the electro-mechanical

commutator, low efficiency, fire hazards due to sparking, limited speed and the extra room

required to house the commutator, became evident. Faraday’s discovery of the concept of

electromagnetic induction in 1831 paved the way towards the invention of induction motors. In

1883 the Serbian-American engineer Nikolai Tesla invented the first alternating-current

induction motor [3]. Tesla’s motor is generally considered to be the prototype of the modern

electric motor. This was the first brushless motor. The synchronous motor, which is also a

brushless motor, was invented by Tesla as well. By the year 1900 the principles of operation of

synchronous and induction motors were well known. But these motors were not widely used at


that time due to the fact that ac power was not yet commercially available. Flexibility of ac

power led to its initial commercial success even though dc power still cost less at that time. AC

power was easier to produce, distribute, and utilize, as compared to dc power. The fierce

competition between ac and dc power was finally resolved in favor of the ac power by 1890. AC

motors have no commutators, and their speed is only limited by the physical constraints of the

motor. These two advantages led to the wide spread utilization of ac motors in motion control

applications in the following decades.

Induction and synchronous motors utilize the same type of stator. But synchronous motors use

a wound dc field or permanent magnet rotor instead of the wire-wound or squirrel cage rotor of

induction motors. Induction motors can generate torque in a wide range of speed. However,

synchronous motors can only generate torque at the synchronous speed. The synchronous speed

depends on the source frequency. Therefore, the first synchronous motors had to run at speeds of

3600, 1800, 900, … RPM for a line frequency of 60 Hz. The speed of synchronous motors has

to first be increased to synchronous speed by means of an auxiliary motor before the motor can

be used. Introduction of the line-start PMSM in the 1950s provided a solution to this problem.

The rotor of line-start PMSM is made of permanent magnet embedded inside a squirrel-cage

winding. Many induction motors utilize squirrel-cage rotors. Induction of current in the squirrel

cage produces torque at zero or higher speeds the same way torque is generated in induction

motors. Therefore, the line-start PMSM can develop torque at zero speed, and run as an

induction motor, until the synchronous speed is reached. Once the rated speed is reached the

rotor is synchronized with the power source, and no more current is induced in the squirrel cage.

After this, the motor runs as a synchronous motor. However, the high cost of line start PMSM

inhibited its wide spread usage. Eventually, a motor drive was used to convert dc power into ac

power with any desired frequency, and to deliver the power to the motor in a controlled manner.

This development allowed the PMSM to be used efficiently at any speed. The introduction of

high performance motor drives has rendered the line start PMSM almost obsolete. Another type

of synchronous motor is the permanent magnet brushless dc motor (BDCM). The rotor of a

BDCM is similar to that of a PMSM. But, its stator is made of concentrated windings. The

distribution of the stator windings of a PMSM is sinusoidal. The developments that have

contributed to the contemporary BDCM can be traced back to the 1960s when dc motors with


permanent magnets began to displace conventional wound field dc motors. The creation of the

rotor field in wound-field dc motors required a separate supply of dc current. This was a major

burden, specially in the case of low power applications where the cost of providing additional

current for the field hindered their commercial use. In the 1960s permanent magnets were

utilized instead of the wound field, which made this motor viable for many servo applications

and other low power motion control systems. However, permanent magnet dc motors still had

the fundamental problems of dc motors mentioned earlier. Eventually, with the advent of

electronic drives, it became possible to place the magnets of the motor on the rotor, and the

windings on the stationary stator. In this case the drive’s primary function is to switch current

into the right phase depending on rotor position. The BDCM motor is also known as the inside-

out dc motor! The BDCM motor does not require a commutator or brushes, since the function of

the brushes is delegated to the drive. Obviously, no sparks are generated because the

commutation of current is not performed mechanically. From a fundamental point of view the

BDCM motor can be considered a special type of PMSM. In both cases the speed of the motor

is proportional to the input current frequency. And, in both cases the magnet is on the rotor and

the winding is on the stator. This makes the cooling of the motor easier as compared to the dc or

induction motor. While the back emf of a PMSM is sinusoidal, the back emf of a BDCM is

trapezoidal. BDCM is usually used for lower performance applications, such as pumps and fans,

and PMSM is mostly used in high performance applications that require high quality torque. It is

only for such applications that the higher cost of a PMSM can be justified.

The torque vs. current relationship for synchronous motor is non-linear. The torque of

synchronous motor depends on both current magnitude and the angle of current phasor with

respect to the rotor. This results in complications as far as control is concerned. Blondel’s study

of synchronous motors [4], 1913, along with Park’s transformation [5], 1933, paved the way

towards linear and instantaneous control over torque for PMSM [6]. Park’s theory presented a

transformation between variables in the stationary and the rotor reference frames which yields

the two-axis equivalent circuit for a PMSM. From the rotor’s point of view every variable has a

magnitude and angle which is constant in steady state. This means that using equations and

variables in the rotor reference frame makes the analysis and control of PMSM much easier.

Essentially, Park introduced auxiliary variables in terms of which the machine equations become


much simpler. Availability of this transformation led to a field referred to as “vector control.”

Vector control enables independent control over the magnitude and angle of current with respect

to the rotor such that instantaneous control over torque is possible. Application of vector control

to a PMSM allows for linear control over torque, as well as control over different performance

criteria such as efficiency and power factor.

Several trends, starting in the 1970s, created a demand for high performance variable speed

motor drives. The trends that are directly relevant to this research effort are discussed next.

2.3 TRENDS FORCING THE TRANSITION TO VARIABLE SPEED DRIVES

The sudden rise in the cost of energy in the 1970s, recent trends towards conservation of the

environment, and the requirements of new applications, have created a significant demand for

variable speed drives. The transition from single speed drives to variable speed drives has left

some areas open for research. These areas include the operational boundary of motor drives and

high performance control strategies for wide speed range motor drives. The issues that deserve

further research in these areas are discussed here in light of the relevant trends during the last

three decades.

Low energy costs in the 1950-70 era resulted in the wide spread use of several types of motors

without regard to efficiency or other performance criteria. However, increasing energy costs in

that last 30 years, public concern over unnecessary use of power and environmental concerns are

driving manufactures to develop more efficient motion control systems. It is estimated that 10%

of all electrical energy is wasted since many motors that do not have drives run at idle for long

periods of time. A much higher percentage of energy is lost simply due to the low efficiency of

many motion control systems. Fans, blowers and pumps represent 50% of all drive capacity

today. These applications can benefit significantly from being able to operate at optimal speed

using a drive. Power consumption increases exponentially with speed. Therefore, running

continuously at a lower optimal speed is better than running in the on-off mode at a higher speed.

Most of today’s fans, blowers and pumps still operate in the on-off mode. Also, on-off operation

wears off the motor faster, and therefore, motors with drives can have a longer life. Drives help

ameliorate transients during startup of motors. Motors without drives draw significant


magnitudes of current during startup and during transient operation. Drives solve this problem

elegantly. Variable speed drives can increase system efficiency by 15 to 27 percent [7]. Many

applications, such as the hybrid vehicle and machine tools, require variable speed operation to

begin with. On the other hand several energy acts, such as the energy act of 1992, impose

minimum efficiency on motors and drives. These trends justify the rapid deployment of drives in

many motion control applications. It is estimated that the capital costs of adding drives to

motors are paid pack in relatively short periods of time due to the savings in energy costs.

All of the issues discussed above have resulted in a significant movement towards the

utilization of variable speed drives instead of single speed systems for many motion control

applications. However, while this transition is in progress, some of the control methods of single

speed systems have been inadvertently carried over to new systems. The operational boundary

of variable speed motor drives are being defined by rated current and power of the machine.

But, these limitations are only valid at rated speed. The consequences, as studied in this

dissertation, are under-utilization of the motor as well as placing the motor at risk of excessive

power losses. This is true for all major classes of motors, i.e. induction motors, BDCM and

PMSM, that are now being utilized in most variable speed applications. Some of the torque

control strategies that are applied to existing advanced PMSM drives are, in fact, carried over

from dc motors. For example, in many cases it is assumed that torque and current are

proportional. Therefore, the torque control strategy is usually based on the assumption that the

relationship between current and torque is linear. However, for several types of PMSM, such as

the inset and interior PMSM, current and torque are not proportional. Some researchers have

presented linear torque control techniques that can be applied to all types of PMSM [8, 9]. The

definition of base speed, generally assumed to be the rated speed of the motor, is another area

where clarification is required. The base speed of a motor is defined here as the speed after

which generation of maximum torque requires application of maximum voltage. In particular the

base speed for interior PMSM is significantly influenced by control strategy, maximum available

voltage, and maximum permissible power loss as later shown in Chapter 5. The theoretical

analysis and implementation strategies presented in this dissertation address the above issues.


2.4 WHY CONCENTRATE THIS STUDY ON PMSM?

While the transition from single speed to variable speed systems is in progress, another

transition is in effect within the field of variable speed motor drives. Direct current and

induction motor drives, which have dominated the field until now, are being replaced by PMSM

and BDCM drives for low power applications. Low power is defined here as being less than 10

kW. Some of the applications for motors below 10 kW are in home appliances, electric tools and

small pumps and fans. PMSM and BDCM have the following advantages over dc motors:

• less audible noise

• longer life

• sparkless (no fire hazard)

• higher speed

• higher power density and smaller size

• better heat transfer

PMSM and BDCM have the following advantages over induction motors:

• higher efficiency

• higher power factor

• higher power density for lower than 10 kW applications, resulting in smaller size

• better heat transfer

The above comparison shows that the PMSM and BDCM are superior to the induction motor for

low power applications. The operation of the BDCM and the PMSM is very similar from a

fundamental point of view. Therefore, all the analysis and control strategies developed for the

PMSM readily applies to the BDCM. The above discussion justifies the choice made in this

dissertation. The same techniques developed in this dissertation can be applied to all high

performance motor drives.

2.5 LITERATURE REVIEW

The state of the art in the following areas is discussed in this section:


• Power losses and efficiency

• Operational boundaries of motor drives

• Control strategies for operation below base speed

• Control strategies for operation above base speed

• Parameter sensitivity and parameter insensitive control strategies

2.5.1 Power Losses and Efficiency

The temperature of a machine rises as a function of power losses in the machine, and machines

have operational limits as far as temperature is concerned. Therefore, the operational boundary

of a motor drive depends on how much loss can be tolerated. The main sources of losses are

copper and core losses. Friction and windage also result in losses. The study of losses and

efficiency are closely related since lower losses at the same torque and speed result in a more

efficient machine. Motors with higher efficiency can be relatively smaller. In other words,

higher efficiency directly translates into higher power density. Efficiency at reduced speed is

critical due to the fact that many drives run at 40% to 80% of rated speed most of the time.

The efficiency and power density of PMSM have been studied by several researchers [10, 11,

12]. [10] gives solid comparison results showing that permanent magnet brushless motors

provide higher efficiency, higher power factor and higher power density for lower than 7 kW

applications as compared to induction motors. [11] gives a comparison between the PMSM and

induction motor showing that the product of efficiency and power factor for PMSM is 30% to

40% higher as compared to induction motor in the lower than 7 kW power range. The higher

efficiency of the PMSM translates into lower losses for same power output as compared to

induction motors. Therefore, the size of a PMSM is smaller than an induction motor capable of

delivering the same power. This results in higher power density for PMSM. [12] shows that the

power density of the BDCM is higher than that of the PMSM.

Copper and core losses are the most fundamental and dominant losses in PMSM. Copper losses

are proportional to the square of current. Core losses have been studied by several researchers

[13, 14, 15]. The main sources of core losses are Eddy currents and hysteresis losses. The

induction of current inside the stator core causes Eddy current losses. Hysteresis losses are the


result of continuous variation of flux linkages in the core. Eddy current losses are nearly

proportional to the square of the product of air gap flux linkages and frequency of the flux

variation. Hysteresis losses are nearly proportional to the product of square of flux linkages and

frequency of flux variation. Several research papers [10, 13, 16] have reported net core losses

that are between 20% to 30% of total loss at rated speed and torque for PMSM below 7 kW.

Core losses are obviously negligible at very low speeds. However, as speed increases the share

of power losses due to core losses increases significantly. Several researchers [17, 18, 19, 20,

21, 22] have utilized an electrical model of PMSM that includes a parallel resistance that

accounts for core losses in high performance applications. [19, 20, 21, 22] deal with efficiency

of the synchronous reluctance motor. It can be concluded from the above discussion that both

copper and core losses need to be accounted for in the analysis and control of high performance

of motor drives.

Another portion of losses is due to stray losses. Stray losses are the result of distortion of the

air gap flux by the phase current [23]. Non-uniform distribution of current in the copper also

leads to stray losses [23]. Stray losses are very difficult to estimate. Therefore, these losses are

usually bundled with core losses during modeling or during experimental measurements.

2.5.2 Operational Boundaries of Motor Drives

The number of research papers that directly investigate the subject of operational limits of

PMSM motor drives for variable speed applications is very limited [16, 24, 25, 26, 28]. [25,26]

deal with choosing motor parameters such that the motor is suitable for a given maximum speed

vs. torque envelop. [24, 25] investigate the optimal design of a motor for delivering constant

power in the flux weakening range. Operating limits of PMSM are studied in [27, 28] based on

the constant power criterion. All papers on high performance control of PMSM define the

operational boundaries of any control strategy by limiting current and power to rated values for

the full range of operating speed.

2.5.3 Control Strategies for Operation Below Base Speed

In the lower than base speed operating range one performance criteria can be optimized while

torque linearity is being maintained at the same time. This degree of freedom can be utilized in


implementing different control strategies. The main control strategies for PMSM for the lower

than base speed operating range are:

(a) Zero d-axis current (ZDAC)

(b) Maximum torque per unit current (MTPC)

(c) Maximum efficiency (ME)

(d) Unity power factor (UPF)

(e) Constant mutual flux linkages (CMFL)

The ZDAC control strategy [29, 30] is widely used in the industry as it forces the torque to be

proportional to current magnitude for the PMSM. The basics behind the MTPC control strategy

have been known for several decades. This control strategy was made popular recently by [31].

The MTPC control strategy [31] provides maximum torque for a given current. This, in turn,

minimizes copper losses for a given torque. The MTPC control strategy is utilized in high

performance applications where efficiency is important, and is the one most studied and utilized

control strategy by PMSM motor drive researchers. However, the MTPC control strategy does

not optimize the system for net loss. The UPF control strategy [29] optimizes the system’s Volt-

Ampere requirement by maintaining the power factor at unity. The ME control strategy [17, 18]

minimizes the net loss of the motor at any operating point. This control strategy is particularly

appealing in battery operated motion control systems in order to extend the life of the system.

The CMFL control strategy [29] limits the air gap flux linkages to a known value which is

usually the magnet flux linkages. This is to avoid saturation of the core.

Each of the above mentioned control strategies have their own merits and demerits. [29]

provides a comparison between the ZDAC, UPF and CMFL control strategies from the point of

view of torque per unit current ratio and power factor. The UPF control strategy is shown to

yield a very low torque per unit current ratio. The ZDAC control strategy results in the lowest

power factor among the five control strategies. [32] provides a comparison between the MTPC

and ZDAC for an interior PMSM. This study shows that the MTPC control strategy is superior

in both efficiency and torque per unit current as compared to the ZDAC control strategy. Torque

is limited to rated value in all existing control techniques for the lower than base speed operating

range. This operating range is referred to as the constant torque operating range. It is shown in


Chapter 3 that the maximum torque in the lower than base speed operating range is not a

constant. A thorough comparison of all five control strategies from the point of view of

maximum torque vs. speed profile, power losses, efficiency, torque per unit current, power

factor, voltage requirements and implementation complexity has not been published. Such a

comparison is made in this dissertation in order to provide a sound basis for choosing the optimal

control strategy for a particular motor drive application.

2.5.4 Control Strategies for Operation above Base Speed

The fundamental component of voltage applied to each phase must remain constant in the

higher than base speed operating range. Performance criteria other than torque linearity cannot

be enforced in this range due to the fact that a restriction on voltage is imposed. This range of

operation is also referred to as the flux weakening range. Two control strategies are possible

depending on whether maximum phase voltage is applied or the phase voltage is limited to a

level lower than maximum possible. These control strategies are:

(a) Constant back emf (CBE)

(b) Six-step voltage (SSV)

The CBE control strategy [8, 27, 33] limits the back emf to a value that is lower than the

maximum possible voltage to the phase. By doing this, a voltage margin is retained that can be

used to implement instantaneous control over phase current. Applications that require high

quality of control over torque in the higher than base speed operating range utilize the CBE

control strategy. The SSV control strategy applies maximum possible voltage to the phase. In

this case only the average torque can be controlled, and a relatively higher magnitude of torque

ripple is present. A procedure for studying the performance of an induction motor under the

SSV control strategy is presented in [34]. Performance evaluation of the SSV control strategy

has been presented for BDCM [35, 36].

References [8, 27, 33, 37, 38] provide insight into the high performance control of PMSM in

the flux weakening range. The topics discussed include linear torque control schemes, speed

range, implementation strategies and torque vs. speed profile. The power is limited to rated


power in all these studies. Therefore, this range of operation is also referred to as the constant-

power operating range. It is later shown in this dissertation that maximum power of a motor is

not a constant in variable speed applications operating above base speed. Most of the control

strategies for the flux weakening range of PMSM result is non-linear control over torque. This

happens either because of neglecting the impact of core losses or by erroneous implementation

strategies. The SSV control strategy is widely utilized in many applications [39]. However, a

comprehensive study of the pros and cons of this control strategy has not been performed.

Ref. [38] provides an implementation scheme for a wide speed range controller for a PMSM.

The transition to flux weakening range is implemented by comparing an estimation of phase

voltage with the maximum available voltage, the latter being itself a measured variable.

Therefore, the transition to flux weakening range happens in an orderly fashion even if the bus

voltage varies. The output of the speed proportional-integrator (PI) controller commands the

magnitude of current to be enforced on each phase. The relationship between current and torque

is not linear in inset and interior PMSM. Therefore, the current command vs. torque gain

changes depending on the magnitude of torque in the above mentioned implementation scheme.

This complicates the design of the speed PI controller. Strictly speaking, the best strategy is to

let the output of the speed PI controller control torque directly so that the speed PI controller can

be designed based on the constant gain relating the torque command to the actual torque.

2.5.5 Parameter Sensitivity and Parameter Insensitive Control Strategies

All high performance control strategies for PMSM are based on an electrical model of the

machine. In most cases, the parameters of the machine are assumed to be constant. In reality

machine parameters vary as a function of temperature and current. Phase resistance of a motor

varies with temperature and frequency. An increase of as much as 100% is possible with a

100 Co rise in temperature. The inductance of some types of PMSM vary by as much as 20% of

the rated value. This is specially true for the interior PMSM [40]. The inductance is a function

of current. The nonlinear magnetic properties of the core are the reason behind the variations of

inductance as a function of current. The magnet’s flux density changes as a function of

temperature [41]. A reduction of 20% is possible for a 100 Co rise in the temperature of the

magnet. If the temperature rises beyond a certain level the magnet may permanently loose part


of its flux density. This temperature threshold depends on the particular characteristics of the

magnets. Another parameter that varies in some applications is the dc bus voltage input to the

drive. The variation of the dc bus voltage affects the speed after which flux weakening must be

initiated. Also, the maximum torque in the flux weakening range and the maximum possible

speed both depend on the dc voltage input value. Inductance can be estimated if the inductance

vs. current relationship is accurately known. Bus voltage can be monitored at all times.

Resistance can be estimated by measuring the temperature of the core. The variations of the

magnet flux density can be estimated if the temperature of the magnet is measured. However,

this is a difficult and costly process due to the fact that the rotor is not stationary. An on-line

estimation of these parameters can be incorporated in the model being used to control the

machine. Core losses also impact the model of a PMSM. Failure to account for core losses

results in torque non-linearity.

Ref. [40] provides an example of a high performance PMSM drive where the phase inductance

in the electrical model is a function of current. Ref. [20] has studied vector control of a

synchronous reluctance motor including inductance variations. Ref. [41] evaluates the impact of

temperature on efficiency and torque of a PMSM. References [42] and [43] present parameter

insensitive control strategies that yield speed and torque linearity, respectively, in the presence of

magnet flux linkages variations.

2.6 CONCLUSIONS

A historical account of motor drives with emphasis on PMSM drives is given in this chapter.

The evolution of permanent magnet synchronous motors and motor drives is studied. The

transition from single speed to variable speed control system is emphasized. It is shown that

some areas, such as the operational envelope, evaluation and comparison of control strategies,

and implementation techniques for wide speed range control strategies, need to be further

investigated. The analytical procedures and implementation schemes provided in this

dissertation are applicable to all motor drives, and can be used to optimize any motion control

system. However, it is shown that the PMSM drive is the most likely beneficiary of the different

concepts introduced in this dissertation. The state of the art in the areas of computation of power

losses and efficiency, operational boundaries, high performance control strategies, and parameter

insensitive control strategies for PMSM, is given.

24

CHAPTER 3

Control and Dynamics of Constant Power Loss Based

Operation of Permanent Magnet Synchronous

Motor Drive System

3.1 INTRODUCTION

The operational boundary of an electrical machine is limited by the maximum permissible

power loss for the machine. The control and dynamics of the PMSM drive operating with

constant power loss are proposed in this chapter. The proposed operational strategy is modeled

and analyzed. Its comparison to other control strategies that limit current and power to rated

values demonstrates the superiority of the proposed scheme. The implementation of the proposed

scheme is developed. This has the advantage of retrofitting the present PMSM drives with least

amount of software/hardware effort. The PMSM drives in this case then can use the existing

controllers to implement any control criteria such as the ones described earlier in sections 2.5.3

and 2.5.4. Experimental verification of the proposed constant power loss operational strategy is

provided.

The maximum torque vs. speed envelope for the control strategies in the lower than base speed

range is commonly found by limiting the stator current magnitude to the rated value. In the

higher than base speed range the shaft power is commonly limited to rated value. Current

limiting restricts copper losses but not necessarily the core losses. Similarly, limiting the shaft

power does not limit power losses directly. Limiting current and power to rated values ignores

the thermal robustness of the machine that requires the total loss be constrained to a permissible

value. Rated current and power guarantee acceptable power loss only at rated speed. Therefore,

these simplistic restrictions are only valid for motion control applications requiring operation at

rated speed. Increasingly at present single speed motion control applications are being retrofitted

Chapter 3 Control and Dynamics of CPL Based Operation of PMSM Motor Drive 25

or replaced with variable speed motor drives to increase process efficiency and operational

flexibility. Also for manufacturing cost optimization, the same machine designs are utilized in

vastly different environmental conditions thus necessitating control methods to maintain the

thermal robustness of the machine while extracting the maximum torque over a wide speed

range. The constant power loss (CPL) based operation provides the maximum torque vs. speed

envelope from these viewpoints, and such a scheme is proposed in this chapter. A comparison of

this operational boundary and the operational boundary resulting from limiting current and power

to rated values is presented. This comparison clearly reveals that the proposed method results in

a significant increase in permissible torque at lower than rated speeds. Consequently, the

dynamic response is enhanced below the base speed. It is also demonstrated that the

conventional method of limiting current and power to rated values can lead to the generation of

excessive power losses in the flux weakening range.

An implementation strategy for the proposed scheme is developed. This implementation

strategy is based on an outer power loss feedback control loop. The input to the system is the

desired maximum power loss of the machine. The feedback loop limits the torque command

such that the power loss does not exceed the maximum set value at any operating point. This

system is applicable to all types of motor drives for the full range of operation, and is

independent of the choice of control strategy for dynamic control of torque. The proposed

implementation strategy can be integrated into all high performance motor drives with very little

modification in their control algorithms. Real-time estimations of machine parameters can also

be utilized as extra inputs to this implementation strategy. Experimental results from a laboratory

prototype are included to verify the proposed implementation to enforce the maximum torque vs.

speed envelope in a torque controlled PMSM drive system. Load duty cycle can be integrated in

maintaining the effective total power loss by varying the power loss reference in the outer control

loop as a function of load duty cycle. Issues concerning load duty cycle are dealt with in Chapter

4. Further it is noted that the underlying control algorithm lends itself to real-time

implementation. The maximum permissible power loss of a machine must be chosen based on a

preset temperature rise for the machine. Therefore, the permissible constant power loss may vary

significantly depending on the operational environment, ambient temperature and the cooling

arrangement for the machine.


All derivations and examples given in this chapter apply to the PMSM. This choice is made

since the PMSM is regularly utilized in high performance applications and can benefit

significantly from the proposed constant power loss controller (CPLC). A similar approach can

be applied to other types of machines with minor modifications.

The contributions of this chapter are summarized below:

• A constant power loss based control strategy to obtain the maximum torque vs. speed

envelope


• An implementation scheme for the proposed CPLC and its flexibility for incorporation in

existing drives that may have various control scheme realizations in its torque and flux

controllers

The CPLC scheme for PMSM is studied using a PMSM model given in section 3.2. The

analysis of the CPLC scheme and its comparison to the control schemes limiting the current and

power to rated values, is presented in section 3.3. The current rating of the drive and parameter

sensitivity of the CPLC scheme are discussed in section 3.4. The implementation strategy for the

CPLC is presented in section 3.5. Experimental correlation is provided in section 3.6.

Techniques for using the power loss command to control the drive are discussed in section 3.7.

The conclusions of this study are summarized in section 3.8. The equations that describe the

constant power loss operational envelope are omitted from this chapter for the sake of clarity.

These equations are provided in Chapter 8 in normalized format.

3.2 PMSM MODEL WITH LOSSES

A dq model for a PMSM in rotor reference frame in steady state with simplified loss

representation is given in Fig. 3.1, where, qsI and dsI are q and d axis stator currents,

respectively, and qsV and dsV are q and d axis stator voltages, respectively. qI and dI are q

and d axis torque generating currents, respectively, and qcI and dcI are q and d axis core loss

currents, respectively. sR and cR are stator and core loss resistors, respectively, and qL and


dL are q and d axis self inductances, respectively. afλ is magnet flux linkages, and rω is the

rotor’s electrical speed.

qsV

dsV

)IL( ddafr +λω

qqr ILω−

qsI

dsI

qI

dI

qcI

dcI

sR

sR

cR

cR

Fig. 3.1. q and d axis steady-state model in rotor reference frame including stator and core loss

resistances.

3.2.1 Electrical Equations of PMSM Including Core Losses

Equations (3.1) and (3.2) are derived from the model of Fig. 3.1.

ωλ+

ω−

ω

=

0

RI

I

1R

L

R

L1

I

Ic

raf

d

q

c

rq

c

rd

ds

qs(3.1)

+λω+

+ω−

+ω=

0

)R

R1(

I

I

R)R

R1(L

)R

R1(LR

V

Vc

safr

d

q

sc

sqr

c

sdrs

ds

qs(3.2)

The torque, eT , as a function of qI and dI is given below,

)II)LL(I(P75.0T qdqdqafe −+λ= (3.3)


where eT is torque, and P is the number of rotor poles.

3.2.2 Total Power Loss Equation for PMSM

The net core loss, lcP , for the machine is computed as follows:

c

2ddaf

2r

c

2qq

2r

lc R

)IL(5.1

R

)IL(5.1P

+λω+ω

= 2m

2r

cR

5.1 λω= (3.4)

where mλ is the air gap flux linkages. Note that in practice a more complex representation of

core losses, based on elaborate equations or tables, can be used to increase the accuracy of the

core loss estimation at higher speeds.

The total machine power loss, lP , including both copper and core losses, can be described as,

])IL()IL[(R

5.1)II(R5.1P 2

ddaf2

qq2r

c

2ds

2qssl +λ+ω++= (3.5)

Another major source of power losses is the electronic inverter. However, inverter losses do not

impact the operational envelope of the machine. This is due to the fact that the cooling

arrangement for the inverter is separate from the cooling arrangement for the machine.

Therefore, inverter losses are not considered in this dissertation. Inverters limit the maximum

current and voltage that can be delivered to a machine. It is presumed that the operating

envelope of the inverter satisfies the motor operation.

In the next section the operational envelope resulting from the application of the CPLC to a

PMSM is discussed.

3.3 CONSTANT POWER LOSS CONTROL SCHEME AND COMPARISON

The maximum permissible power loss, lmP , depends on the desired temperature rise for the

machine. lmP can be chosen to be equal to the net loss at rated torque and speed assuming that


the machine is running under exact operating conditions defined in manufacturer data sheets. At

any given speed the current phasor, which is the resultant of qI and dI , and its trajectory for

maximum power loss is given by (3.5) with lP replaced by lmP . This trajectory is a circle at zero

speed, and a semi-circle at non-zero speeds. The operating point of a PMSM must always be on

or inside the trajectory defined by (3.5) for that speed so that the net loss does not exceed lmP .

At any given speed the operating point on the constant power loss trajectory which also results in

maximum torque defines the operational boundary at that speed. At this operating point,

maximum torque is generated for the given power loss of lmP . To find the maximum

permissible torque at a given speed it is sufficient to move along the trajectory defined by (3.5)

for a given lmP and find the operating point that maximizes (3.3). In the flux weakening range,

both voltage and power loss restrictions limit the maximum torque at any given speed. The

following relationship is true for any stator current phasor operating point in the flux weakening

range assuming that the voltage drop across the phase resistance is negligible,

mrr5.02

ddaf2

qqsm ])IL()IL[(V λω=ω+λ+= (3.6)

where smV is either the maximum desired back emf or the fundamental component of maximum

voltage available to the phase. The latter applies to the six step voltage control strategy. At any

given speed in the flux weakening range the stator current phasor that results in a set power loss

can be found by solving equations (3.5) and (3.6). This operating point corresponds to the

maximum permissible torque at the given speed in the flux weakening range. Fig. 3.2 shows the

maximum torque possible for the full range of speed for the motor drive described in Appendix I.

All variables in Fig. 3.2 are normalized using rated values. The power loss is limited to the rated

value of 121 Watts at all operating points. The following assumption is used in solving the

required equations as discussed above,


Fig. 3.2. Normalized maximum torque, power loss, air gap power, voltage and phase current vs.

speed for the CPLC (solid lines) and for the scheme with current and power limited to rated

values (dashed lines).

p.u.,Ten

p.u.,Pln

p.u.,Pan

p.u.,Vsn

p.u.,Isn

p.u.,rnω

p.u.,Ten

p.u.,Pln

p.u.,Pan

p.u.,Vsn

p.u.,Isn

p.u.,rnω

p.u.,Ten

p.u.,Pln

p.u.,Pan

p.u.,Vsn

p.u.,Isn

p.u.,rnω


2d

2q

2ds

2qs IIII +=+ (3.7)

The slight deviation of the power loss from the set value of 1 p.u. in Fig. 3.2 is due to this

assumption.

It is to be noted that the operating points along the operational envelope described here are also

the most efficient operating points. Therefore, only the maximum efficiency control strategy for

torque can lead to an operating point along this envelope. Other control strategies, such as

maximum torque per current and constant torque angle, result in operational envelopes that are

smaller and narrower than that resulting from the maximum efficiency control strategy. The

comparison of different control strategies under the constant power loss concept is addressed in

Chapter 5.

3.3.1 The Lower than Base Speed Operating Range

The base speed, bω , is defined as the speed beyond which the applied phase voltage must

remain constant along the CPLC scheme boundary. In the lower than base speed operating range

torque is only limited by the power loss, while phase voltage is less than maximum possible

value, smV . This range of operation is shown in Fig. 3.2 between 0 and 1.1 p.u. speed. Power

loss, air gap power, phase voltage and current along the CPLC scheme boundary are also shown

in Fig. 3.2.

3.3.2 The Flux Weakening Operating Range

The portion in Fig. 3.2 between 1.1 and 1.55 p.u. speed corresponds to the flux weakening

range of operation, where back emf is limited to 1.1. p.u. It is seen that the CPLC boundary

drops at a faster rate in this range due to voltage restrictions. The maximum possible air gap

power continues to rise beyond rated speed up to approximately 1.25 p.u. speed.


The operational boundary resulting from limiting current and power to rated values is also

shown in Fig. 3.2. It can be concluded from the example that the application of the constant

current and power operational envelope results in:

• Under-utilization of the machine at lower than base speed

• Generation of excessive power losses at higher than rated speeds unless both power and

current are limited to rated values in the flux weakening range

• Under-utilization of the machine in some intervals of the flux weakening range

3.4 SECONDRY ISSUES OF THE CPL CONTROLLER

3.4.1 Higher Current Requirement at Lower than Base Speed

It is seen from Fig. 3.2 that the CPLC provides 39% higher torque at zero speed as compared to

the maximum torque possible with rated current. This requires 36% more than rated current at

zero speed or in the very low speed operating range. It is to be noted that this additional current

requirement does not result in a proportional increase in the price of the drive. This is mainly

due to the fact that the voltage requirements of both control strategies are similar as seen from

Fig. 3.2. Therefore, the power switches have to be upgraded only for current and not for higher

voltage.

3.4.2 Parameter Dependency

The CPLC scheme is dependent on machine parameters dL , qL , afλ , cR and sR . The d-axis

path of the rotor involves a relatively large effective air gap, and does not saturate under normal

operating conditions. Therefore, dL does not vary significantly. qL varies as a result of

magnetic saturation along the q-axis, and can be estimated accurately as a function of phase

current [40]. An accurate estimation of afλ requires more complex algorithms [43]. sR varies

as a function of temperature. Any implementation strategy for the CPLC is by nature parameter-

dependent. This is the case with all model-based control strategies.


Motor

mθ

*eT

~

mω

+

_

Wide speed-range lineartorque controller

Resolver &Tachometer

*qsi

*dsi

Current controllerand power stage

Power lossestimator

*eT

+ _*lP

limT

mω

lP

Peaktorquelimiter

PI Controller Filter

PI

*rω

mω

lfP

Fig. 3.3. Implementation scheme for the constant power loss controller.

3.5 IMPLEMENTATION SCHEME FOR THE CPL CONTROLLER

Fig. 3.3 shows the block diagram for an implementation strategy of the CPLC scheme. The

wide speed-range linear torque controller is assumed to provide torque linearity over the full

range of operating speed including the flux weakening range. Any control strategy can be

utilized in the torque controller block. The copper and core losses of the machine are estimated

using (3.5) utilizing the q and d axis current commands, *qsi and *

dsi , as well as measured speed,

mω . All the required variables for power loss estimation are already available within most high

performance control systems. The estimated net power loss, lP , is always a positive number as

seen from (3.5). lfP is the filtered version of lP . The filtered power loss estimation is

compared with the power loss reference, *lP . The difference is processed through a proportional

and integrator (PI) controller. The output of the power loss controller determines the maximum

permissible torque, limT . The torque command, *eT , is the output of a PI controller operating on

the difference of the commanded and measured speeds. If the torque command is higher than

limT then the system automatically adjusts the torque to the maximum possible value, limT .


However, if the torque command is less than the maximum possible torque at a given speed then

the torque limit is set at absolute maximum value leaving the torque command unaltered. The

same absolute value of the torque limit is applied to both positive and negative torque

commands. The magnitude of transient torque is limited by the peak torque limiter block. The

processed torque command, *eT

~, and mω , are inputs to the wide speed range linear torque

controller. The outputs of this block are *qsi and *

dsi . The current controller and power stage

enforce the desired current magnitude and its phase on the motor. The inputs to this stage are

*qsi , *

dsi , and rotor position, mθ .

The salient features of the CPL controller are summarized below:

• Off-line calculation of the maximum torque vs. speed envelope is not necessary

• Maximum power loss can be adjusted by an operator or by process demand

• An on-line estimation of qL , afλ and sR can be used by the power loss estimation block to

increase accuracy

• The system is independent of the control strategy used in the linear torque controller block

• All necessary parameters are already available in most high performance controllers

• The scheme lends itself to real-time implementation

3.6 EXPERIMENTAL CORRELATION

Experimental correlation is provided using a prototype motor drive utilizing an interior PMSM

with parameters given in Appendix I. The experimental PMSM motor drive is set up as a torque

controller for lower than base speed operating range. The power loss controller is implemented

as shown in Fig. 3.3. The functions required in estimating the machine power losses are

implemented using 8 bit multiplying DACs. The maximum efficiency control strategy is

implemented for the dynamic control of torque. This requires that the q and d-axis currents be

functions of both measured speed and torque commands. Two 8 kByte EEPROM lookup tables

are utilized in this process. The maximum set power loss is 30 W. Fig. 3.4 shows the maximum


possible measured torque and current, predicted maximum torque, as well as corresponding core,

copper and net losses vs. speed. The torque command is set at the maximum value of 1.5 N.m.

It is seen that as speed increases the maximum possible torque and current decline monotonically

while core losses increase and copper losses decrease. However, the net loss remains fairly close

to the desired set value of 30 W. The measured maximum torque vs. speed envelope is very

close to the estimated maximum torque vs. speed envelope as seen from Fig. 3.4. The slight

difference is due to torque measurement error. Fig. 3.5 shows measured speed and phase current

command of the system in response to a step torque command of 1.5 N.m. It is seen that as

speed increases the phase current magnitude decreases automatically, resulting in the reduction of

the maximum possible torque to 1.03 N.m. at 2,500 RPM in order to maintain the net loss at 30

W. Speed is limited primarily by the bus voltage in this case. This experiment demonstrates the

robustness of the proposed implementation scheme under dynamic conditions. In practice, the

maximum power loss reference can be modified dynamically as a function of load duty cycle or

based on process demand. Fig. 3.6 shows a simulation of torque step response of the CPLC

system as described above. It is seen that the predicted speed and phase current command

closely correlated to the experimental result given in Fig. 3.5.

3.7 THE DRIVE CONTROL WITH POWER LOSS COMMAND

3.7.1 Cycling the Drive

The power loss command, *lP , can be used to stop and start the motor drive. Commanding *

lP

to be equal to zero effectively turns off the drive by driving the torque limit to zero. Any nonzero

power loss command allows torque to be applied to the motor.


Measured current, A.

Estimated torque, N.m.

Measured torque, N.m.

Total power losses, W.

Copper losses, W.

Core losses, W.

Speed, RPM

Fig.3.4. Current, torque, estimated torque, total power loss, copper and core losses along the

CPL boundary with power loss reference of 30 W.

3.7.2 Commanding Short Term Maximum Torque

*lP can also be used to command peak torque values for a short time by commanding

maximum power loss. The magnitude of the peak torque is set by the peak torque limiter block

as seen in Fig. 3.3. However, the duration of a higher than nominal power loss command must

not be long enough to damage the motor. The absolute torque limit must be set to a reasonable

value as part of the installation procedure of the motor drive.


Time, 100 ms/div.

Speed, 1000 RPM/div.

Phase current command, 2 A/div.

Fig. 3.5. Dynamic response of the system for a step torque command of 1.5 N.m.

Time, s

mω

*ai

Fig. 3.6. Simulated speed, mω (RPM), and phase current command, *ai (A), in response to 1.5

N.m. step torque command.


3.8 CONCLUSIONS

A new control scheme for PMSM based on constant power loss is proposed and analyzed. It is

shown that the constant power loss operation maximizes the utilization of the machine in wide

speed range applications. A comparison of the proposed operational envelope with the

operational envelope resulting from limiting current and power to rated values is performed. It is

shown that the proposed scheme results in higher than rated torque at lower than base speed, as

well as higher than rated power in the vicinity of base speed. Overall, the proposed scheme

results in better dynamic response at lower than base speed and thermal robustness of the

machine at all operating points. It is also shown that maintaining rated power in the higher than

base speed may compromise the thermal stability of the machine. An implementation strategy for

the proposed control scheme is developed. This implementation strategy limits the torque to

correspond to the chosen power loss of the machine by employing an outer feedback loop to

control machine power losses. This has the advantage of retrofitting the present PMSM drives

with least amount of software or hardware effort. The implementation strategy does not require

any off line calculation, and is independent of the chosen control strategy for the dynamic control

of torque. The scheme lends itself to on-line implementation. The maximum permissible power

loss is the main input to the system. However, an on-line estimation of other parameters can be

input to the system in order to increase the accuracy of the power loss control system.

Experimental correlation confirms the key results of this chapter.


• A constant power loss based control strategy to obtain the maximum torque vs. speed

envelope


• An implementation scheme for the proposed CPLC and its flexibility for incorporation in

existing drives that may have various control scheme realizations in its torque and flux

controllers

• Experimental verification of the proposed scheme

39

CHAPTER 4

The Constant Power Loss Controller for Applicationswith Cyclic Loads

4.1 INTRODUCTION

Many motion control applications require cyclic accelerations, decelerations, stops and starts.

The required movements are usually programmed into the system using a microcontroller. A

simple programmed move is the cyclic on-off operation with negligible fall and rise times. The

load duty cycle, in this case, is defined as the ratio of on time to the cycle period. In many

applications speed is constant during the on time. Usually several times average torque is

required during transitions from zero to maximum speed and vice versa. Different power losses

are generated during different phases of operation of a motor drive with cyclic loads. For

applications where the cycle period is significantly smaller than the thermal time constant of the

machine the average power loss during one cycle must be limited to the maximum permissible

value. This type of application is dealt with in this chapter. Generally speaking, the maximum

possible torque increases as the duty cycle decreases. This is due to the fact that no power loss is

generated during the off time. It is to be noted that in some applications the cycle period is large

compared to the thermal time constant of the machine. In these applications the average power

loss during one cycle in not of significant value, and instead, the instantaneous power loss during

on time must be limited to the maximum permissible value.

The objective of this chapter is to calculate the appropriate power loss command for a constant

power loss controller applied to different applications with cyclic loads. It is assumed that the

maximum permissible power loss, lmP , during continuous operation of the machine at steady

state, is known. It is also assumed that the on and off times are significantly shorter than the

thermal time constant of the machine. Five different categories of applications with cyclic loads

are considered here. In each case the power loss command must be calculated such that the

average loss over one period is equal to the maximum permissible power loss for the machine.

Chapter 4 The CPL Controller for Applications with Cyclic Loads 40

The power loss command and maximum possible torque are calculated in each case as a

function of duty cycle, maximum permissible power loss and maximum speed during one cycle.

The procedures discussed in this chapter can be applied to any application with cyclic loads.

In section 4.2 major motor drive applications are classified into five categories as far as cyclic

loads are concerned. In each category the required power loss command, average power loss,

and maximum torque during on-time, are calculated. A comparison of the maximum possible

torque vs. maximum power loss for different load categories is given in section 4.3. The

conclusions are summarized in section 4.4.


• Calculation of the appropriate power loss commands for applications with cyclic loads

• Derivation of average power loss as a function of duty cycle, maximum speed and torque for

applications with cyclic loads

• Derivation of maximum possible torque as a function of load duty cycle, maximum power

loss and maximum speed for applications with cyclic loads

• Comparison of maximum possible torque as a function of maximum possible power losses

for different applications with cyclic loads

4.2 THE POWER LOSS COMMAND IN DIFFERENTAPPLICATION CATEGORIES

A simple and practical power loss estimator for PMSM is developed here. This power loss

estimator is used in calculating the appropriate power loss command, average power loss, and

maximum possible torque, in each of five different categories of applications with cyclic loads.

Most applications fall in one of these five categories. Applications that are not covered in this

chapter can be treated using similar procedures. All torque and speed profiles are normalized

using maximum possible torque and speed in each category.

4.2.1 Derivation of a Practical Power Loss Estimator

An instantaneous power loss estimator, applicable to the surface mount PMSM, is developed

here. Note that most high performance motion control applications utilize the surface mount

PMSM. All derivations are based on the following assumptions:


• D-axis current is zero

• The difference between air gap and magnet mutual flux linkages is negligible

• Torque and current are proportional

• Impact of core losses on torque linearity is negligible

The above assumptions are very closely valid for the surface mount PMSM. These assumptions

are also valid for the BDCM. Therefore, the results of this chapter are readily applicable to the

BDCM. However, these assumptions are not valid for inset and interior PMSM unless these

machines are operated using the zero d-axis control strategy. The inset and interior PMSM are

only used in a small percentage of all high performance PMSM applications. Based on the

above assumptions,

ste iKT = (4.1)

where eT is the machine torque, si is the stator current magnitude, and,

aft P75.0K λ= (4.2)

The equation for the instantaneous power loss, lP , for a PMSM, can be derived using (3.5), (4.1)

and (4.2) by applying the assumptions discussed above. lP is given below,

2r2

2e1l KTKP ω+= (4.3)

where,

2t

s1

K

R5.1K = ,

c

2af

2 R

5.1K

λ= (4.4)

Note that the air gap flux linkages and magnet flux linkages are almost equal for the surface

mount PMSM. This fact is used in calculating 2K as given in (4.4).

4.2.2 *lP and emT in Different Application Categories

In this section motion control applications are broadly classified in five categories as far as

cyclic loads are concerned. In each category the required power loss command, *lP , is


calculated. All calculations are based on the implementation scheme described in Fig. 3.3. The

average power loss, lP , and maximum possible torque, emT , as a function of the maximum

permissible power loss, lmP , and for a given maximum speed, rmω , and load duty cycle, d, are

calculated in each category.

A. Continuous Operation at Constant Speed and Torque

Applications such as conveyers, escalators, some types of fans/pumps/compressors, electric

vehicles in cruise mode, and disk drives fall in this category. The speed and torque remain

essentially constant during the operation. The load duty cycle is unity in this category. Fig. 4.1

shows the torque and speed profiles for a typical application in this category. The variables are

normalized using maximum possible torque, emT , and maximum possible speed, rmω . The

average power loss in this case is,

Time, s

em

eT

T

rm

rωω

(p.u.)

(p.u.)

Fig. 4.1. Normalized torque, eT , and speed, rω , profiles for continuous operation.


ll PP = (4.5)

where lP is the instantaneous power loss of the machine as described in (4.3). On the other

hand, the maximum average power loss is defined below,

lml PP = (4.6)

It can be seen from (4.5) and (4.6) that the maximum instantaneous power loss in steady state is

constant in this category, and is given below,

lml PP = (4.7)

Therefore, the maximum possible power loss command in this category is,

lm*l PP = (4.8)

It can be concluded from (4.3) and (4.7) that,

2rm2

2em1lm KTKP ω+= (4.9)

Therefore, the maximum possible torque, emT , for maximum power loss of lmP is,

5.0

1

2rm2lm

em )KKP

(Tω−= (4.10)

B. On-off Operation with Negligible Rise and Fall Times

This category applies to applications that run periodically in on-off mode but have negligible

rise and fall times compared to the cycle period. Some air conditioning and

fan/pump/compressor applications fall in this category. Fig. 4.2 shows an example of a torque

and speed profile in this category. In this category the operating point of the machine rises to the

desired point rapidly. Then the operating point stays constant for onT seconds. Subsequently,

the operating point drops to zero rapidly. The machine remains at this operating point for offT

seconds. The net power loss during rise and fall of the operating point is negligible as compared


to the losses during on time. These applications may require higher than rated torque during the

short rise and fall periods.

em

eTT

rm

rωω

Time, s

(p.u.)

(p.u.)

onT offT

Fig. 4.2. Normalized torque and speed profiles for on-off operation with small transition times.

The average power loss in this case is,

dPTT

TPP l

offon

onll =

+= (4.11)

where lP is the power loss during on time. On the other hand, the maximum average power loss

is defined below,

lml PP = (4.12)

It can be concluded from (4.11) and (4.12) that, while operating at maximum average power loss,


lml PdP = (4.13)

Therefore, the maximum power loss command during on time is defined below,

dP

P lm*l = (4.14)

The maximum torque during on time, emT , can be calculated by substituting (4.3) into (4.13) as

given below:

lm2rm2

2em1 Pd)KTK( =ω+ (4.15)

5.0

1

2rm2lm

em )K

KdP(T

ω−= (4.16)

C. Speed Reversal Operation with Negligible Transition Times

In some motion control systems, such as pick and place applications and automatic welding

machines, the end-effector is moving in between jobs most of the time while the actual task takes

a relatively short time for completion. Therefore, the power loss during the actual task time is

negligible compared to the average power loss during one full cycle. The speed of the end-

effector is almost constant while moving in between jobs. Fig. 4.3. shows a typical torque and

speed profile of an application in this category. As far as the net power loss is concerned this

profile and the profile described in section A are the same. Therefore, the maximum power loss

command can be calculated, using (4.8), as given below,

lm*l PP = (4.17)

Similarly, the average power loss, under extreme conditions, is described as,

lml PP = (4.18)

The maximum possible torque, emT , for maximum power loss of lmP and speed of rmω , is,


Time, s

em

eTT

rm

rωω

(p.u.)

(p.u.)

Fig. 4.3. Normalized torque and speed profiles for speed reversal application.

5.0

1

2rm2lm

em )KKP

(Tω−= (4.19)

D. Speed Varying Linearly Between rmω±

Applications, such as industrial robots and some pick and place machines, where the end

effector is always being accelerated in positive or negative directions, fall in this category. Fig.

4.4 shows an example of the torque and speed profile of an application in this category. The

average power loss during the first half of one period is equal to the average power loss during

the second half.


(p.u.)

(p.u.)

em

eTT

rm

rωω

Time, sT

Fig. 4.4. Normalized torque and speed profiles for operation between rmω± .

The average power loss in the first half of one period can be calculated as,

T

dtKTKP

T0

2r22

em1l∫ ω

+= (4.20)

where,

rmrm

r t)T

2( ω−ω=ω (4.21)

and T is one half of a cycle period. In this case (4.20) can be simplified as,

2rm2

2em1l K

31

TKP ω+= (4.22)


In this category the instantaneous power loss estimation changes with speed throughout every

period. If the power loss controller’s PI block operates on the difference of the power loss

command and the instantaneous power loss estimation, then the resulting torque limit varies

linearly within one cycle of operation. However, as seen in Fig. 4.4, the torque needs to be

constant during each half period of one full cycle. The solution to this problem is to use a low

pass filter with a large time constant on the output of the power loss estimator (see Fig. 3.3).

Such a filter should effectively output the average power loss of the machine in steady state.

Under these conditions the maximum power loss command, that can limit the average power loss

to lmP , can be described as,

lm*l PP = (4.23)

Since the maximum average power loss must not exceed lmP , it can be concluded from (4.22)

that,

lm2rm2

2em1 PK

31

TK =ω+ (4.24)

The maximum possible torque can then be calculated from (4.24) as,

5.0

1

2rm2lm

em )K

KP3(T

ω−= (4.25)

E. On-off Operation with Significant Rise and Fall Times

Industrial lifts, elevators, and some pumps, fans and servo drives are examples that fall in this

category. For these applications the speed rises to a target level while several times rated torque

is being applied. Speed and torque remain constant for a period of time. Then the speed is

reduced to zero which again requires several times rated torque. The operating point stays in this

state for a period of time. The power losses during rise and fall times, in this category, constitute

a significant portion of the average power loss.


Time, s

rm

rωω

em

eTT

em

ep

T

T

em

ep

T

T−

1pT∆ mT∆

2pT∆zT∆

(p.u.)

(p.u.)

Fig. 4.5. Normalized torque and speed profiles for operation with significant transition times.

Fig. 4.5 shows a typical torque and speed profile in this category. A peak torque of epT N.m. is

applied for 1pT∆ seconds at the beginning of each cycle to raise the speed to the desired value of

rmω . Then speed is maintained constant for a period of mT∆ seconds during which time a

constant torque of emT N.m. is applied to the load. Finally, speed is brought back to zero by

applying epT− N.m. for a period of 2pT∆ seconds. The speed remains at zero for a period of

zT∆ seconds.

The magnitude of peak torque is set by the peak torque limiter block of Fig. 3.3. The power

loss command, *lP , is set at maximum value during the peak torque periods in order to saturate

the power loss PI controller. Saturation of this PI controller allows the torque to be limited by


the peak torque limiter block of Fig. 3.3. The power loss command during the mT∆ seconds

where nominal torque is being applied must be calculated as follows. The average power loss in

this case is calculated by adding the energy losses in each segment, and dividing the result by the

time period of one cycle. The net energy loss, lW , in one cycle can be described as,

2p2rm2

2ep1m

*l1p

2rm2

2ep1l T)K

31

TK(TPT)K31

TK(W ∆ω++∆+∆ω+= (4.26)

The average power loss in one cycle is,

T

W

TTTT

WP l

z2pm1p

ll =

∆+∆+∆+∆= (4.27)

where T is the cycle period. The average power loss should not exceed lmP . Therefore,

lml PP = (4.28)

It can be concluded from (4.26), (4.27) and (4.28) that

2p2rm2

2ep1m

*l1p

2rm2

2ep1lm T)K

31

TK(TPT)K31

TK(TP ∆ω++∆+∆ω+= (4.29)

Therefore, the required *lP during the mT∆ period can be calculated from (4.29) as,

m

2p2rm2

2ep11p

2rm2

2ep1lm*

l T

T)K31

TK(T)K31

TK(TPP

∆

∆ω+−∆ω+−= (4.30)

In brief, the power loss command is set at maximum during the transitional periods, and is set at

the value calculated in (4.30) for the steady state period.

The power loss during the steady state period is,

)KTK(P 2rm2

2em1l ω+= (4.31)


where lP is calculated from (4.30), and is equal to *lP in steady state. Therefore, the maximum

possible torque during the mT∆ period in Fig. 4.5 is,

5.0

1

2rm2

*l

em )KKP

(Tω−= (4.32)

An alternative to the CPL implementation scheme described in Chapter 3 is to program the

torque limiter to limit the torque to maximum desirable values during transition periods, and to

limit the torque the value given in (4.32) during nominal operation. In this case the power loss

control loop is not required. Similarly, the torque limiter can be utilized in limiting the average

power loss in other application categories without resorting to the power loss feedback control

loop. However, having a power loss control loop is preferred in the first four categories because

of the simplicity of its implementation and the simplicity involved in calculating the power loss

command in those categories.

4.3 COMPARISON OF MAXIMUM TORQUE IN DIFFERENT CATEGORIES

It can be concluded from the previous discussion that the maximum possible torque is a

function of maximum possible power loss, load duty cycle and the nature of the cyclic load. In

this section the maximum possible torque as a function of lmP and load duty cycle is calculated

and compared for the load categories A to E as discussed in sections 4.2.a to 4.2.e. The

parameters of the motor drive given in Appendix I are used for this purpose. It is assumed that

the zero d-axis current control strategy is adopted so that the assumptions given in section 4.2.1

are valid. The maximum speed is set at rated speed in all cases. The load categories A to D are

dealt with in section 4.3.1. The load category E is dealt with separately in section 4.3.2. due the

large number of variables that can affect the maximum possible torque in that category.

4.3.1 Load Categories A to D

Figure 4.6. shows the maximum torque vs. maximum power loss profiles for load categories A

to D. The maximum torque vs. maximum power losses equations derived in sections 4.2a to

4.2.d have been used in preparing the profiles for load categories A to D, respectively.


Fig. 4.6. Maximum torque vs. maximum power loss for load categories A, B, C and D for duty

cycles equal to 1, 0.75, 0.5, 0.33.

Load categories A and C result in identical maximum torque vs. maximum power loss profiles.

The maximum torque is the lowest in these categories due to the lack of any off time. Load

category B with load duty cycle of 1 also results in the same profile as that of load categories A

and C. However, as duty cycle is lowered the maximum possible torque for a given maximum

power loss consistently increases. The torque vs. power loss profile for load category D is the

same as that of load category B with a duty cycle of 0.333. This can be seen by comparing

(4.25) and (4.16). In each load category there is a minimum power loss below which no torque

can be produced. This is due to the fact that the maximum possible power loss should exceed the

net core losses generated within each period so that a nonzero magnitude of copper losses can be

tolerated. Note that generation of torque is not possible if copper losses are not allowed.

lmP

emT

A , C, or B with d=1

B with d=0.75

B with d=0.5

B with d=0.33, or D

lmP

emT


B with d=0.75

B with d=0.5

B with d=0.33, or D

lmP

emT


B with d=0.75

B with d=0.5

B with d=0.33, or D

lmP

emT


B with d=0.75

B with d=0.5

B with d=0.33, or D

lmP

emT


B with d=0.75

B with d=0.5

B with d=0.33, or D

lmP

emT


B with d=0.75

B with d=0.5

B with d=0.33, or D

p.u.,Plm

p.u.,Tem


B with d=0.75

B with d=0.5

B with d=0.33, or D

p.u.,Plm

p.u.,Tem


B with d=0.75

B with d=0.5

B with d=0.33, or D


4.3.2. Load Category E

The maximum possible torque for applications with significant rise and fall times is a function

of maximum possible power loss, load duty cycle, peak torque requirements and duration of rise

and fall times. In this section a sample calculation of maximum possible torque for such an

application using the prototype motor drive discussed in Appendix I is given. The peak torque is

assumed to be twice the rated torque, and the duration of rise and fall times are 2 seconds each.

The entire period is 20 seconds, and the machine is at idle for 6 seconds. This leaves 10 seconds

during which torque is applied to the application while speed is constant at the target value. The

objective is to find the maximum possible torque during this period. Equation (4.30) yields a

maximum possible power loss of 93.1 W during this period. The maximum possible torque is

0.84 p.u. in this case using (4.32). If the on time is reduced to 8 seconds instead of 10 seconds

then the maximum possible torque rises to 0.98 p.u. An on time of 6 seconds results in a

maximum possible torque of 1.19 p.u. Note that the power losses during the transition times do

not change as duty cycle changes since the maximum torque during transition time is fixed.

It is clear from the above discussion that the calculation of maximum possible torque in the case

of applications with load profiles such as the one shown in Fig. 4.5 is subjective and needs to be

dealt with on case by case basis as shown above.

4.4 CONCLUSIONS

A large number of motion control applications involve cyclic and programmable movements.

The procedure for adapting the constant power loss controller, as described in Chapter 3, to

applications that involve cyclic movements is discussed in this chapter. The maximum

permissible power loss, lmP , for a machine has been previously defined to be valid only for

constant speed and torque operation. In this chapter the power loss command is calculated as a

function of duty cycle, lmP and maximum speed in five different categories of applications with

cyclic loads such that the average power loss does not exceed lmP . These categories are: A.

constant torque and speed, B. on-off operation at constant speed and torque at a specific duty

cycle, C. speed reversal with constant torque and speed magnitudes at a specific duty cycle, D.

cyclic ramping operation between two speeds with constant magnitude of torque, E. general


torque vs. speed profiles. It is shown that in load categories A and C the power loss command is

equal to lmP . In load category B the power loss command is equal to lmP divided by the duty

cycle. In load categories A, B and C the power loss estimator filter can be a simple filter to

remove noise, and should yield the instantaneous net power loss of the machine. The power loss

command in load category D is equal to lmP . It is shown that, in this case, the power loss

estimator filter must yield the average of the net power loss in one cycle instead of the

instantaneous power loss. In load category E each subsection in one cycle has its own maximum

possible torque. Therefore, the power loss command in each subsection is different.

Consequently, the power loss command must be programmed as a function of the desired torque

vs. speed profile in this category. Instantaneous power loss estimation is required in this

category. In each of the five categories the average power loss and maximum possible torque as

a function of lmP , speed and duty cycle are calculated. All load categories are compared based

on maximum possible torque as a function of maximum possible power loss and load duty cycle.

A similar approach, as outlined in this chapter, can be applied to applications that involve more

complex torque and speed profiles.


• Calculation of the appropriate power loss command for applications with cyclic loads

• Derivation of average power loss as a function of duty cycle, maximum speed and torque for

applications with cyclic loads

• Derivation of maximum possible torque as a function of duty cycle, maximum power loss

and maximum speed for applications with cyclic loads

• Comparison of maximum possible torque as a function of maximum possible power losses

for different applications with cyclic loads

55

CHAPTER 5

Performance Evaluation and Comparison of

Control Strategies 5.1 INTRODUCTION

Vector control of PMSM allows for the implementation of several choices of control strategies

while control over torque is retained. The main control strategies for the lower than base speed

operating range are:

(a) Maximum efficiency (ME)

(b) Maximum torque per unit current (MTPC)

( c) Zero d-axis current (ZDAC)

(d) Unity power factor (UPF)

(e) Constant mutual flux linkages (CMFL)

The main control strategies for the higher than base speed operating range are:

(a) Constant back emf (CBE)

(b) Six-step voltage (SSV)

Each of these control strategies are described and analyzed in this chapter. The procedure for

deriving the q and d-axis current commands as a function of torque and speed is described in

each case. Also, the procedure for deriving the maximum possible torque vs. speed profile for a

given maximum possible power loss is described for each control strategy. Subsequently, a

comprehensive comparison of these control strategies is made. The main part of the comparison

is based on operation along the maximum possible torque vs. speed envelope. The maximum

possible torque vs. speed envelope depends on the maximum possible power loss, and is also a

function of the chosen control strategy as well as motor drive parameters. Therefore, each

Chapter 5 Performance Evaluation and Comparison of Control Strategies

56

control strategy results in a unique operational envelope for a given machine. Consequently, the

performance of the system under each control strategy is also unique. Maximum torque,

current, power, torque per unit current, back emf and power factor vs. speed are the key

performance indices that are used here to compare different control strategies. Current, air gap

flux linkages, and d-axis current vs. torque are also compared in each case in order to provide

insight into the performance of the system inside the operational boundary. It is assumed here

that the maximum possible power loss is constant for the full range of operating speed. This

assumption is made in order to simplify the demonstrations and analytical derivations, and to be

able to present the fundamental concepts with better clarity. Similar procedures can be used to

analyze and compare different control strategies for an arbitrary maximum possible power loss

vs. speed profile. The procedures provided in this study can be used to choose the optimal

control strategy based on the requirements of a particular application and also based on the

capabilities of the chosen motor drive. The fundamental concepts and procedures are emphasized

in this chapter, while non-critical analytical derivations are delegated to Chapter 8.


• Performance analysis of different control strategies for PMSM along the constant power loss

operational envelope

• Comparison of control strategies along the constant power loss operating envelope

• Comparison of control strategies inside the operating envelope

• Derivation of absolute maximum torque for the unity power factor and constant mutual flux

linkages control strategies

• Derivation of maximum speed, current and torque in the flux weakening range as a function

of maximum power loss and machine parameters.


Section 5.2 describes a complete set of performance criteria. Section 5.3 reviews the analytical

details of the five control strategies applicable to a high performance PMSM control system in

the lower than base speed operating range. The two possible control strategies for the higher

than base speed operating range are studied in section 5.4. The performances of control


57

strategies along the constant power loss operational envelope are compared in section 5.5. The

performances of control strategies inside the operational envelope are compared in section 5.6.

The procedure used to study torque ripple under the SSV control strategy is described in section

5.7. Simulation and experimental verification of the key results of this chapter are given in

section 5.8. The conclusions are discussed in section 5.9.

5.2 PERFORMANCE CRITERIA

The performance criteria discussed below are the most relevant as far as this study is concerned.

These criteria are used in later sections to evaluate and compare the performances of different

control strategies for the full range of operational speed.

(a) Torque, current and voltage: These three variables are among the most fundamental

performance criteria in motor drives. The basic function of a motor drive is to deliver the torque

required by an application. It is desirable to produce a higher magnitude of torque within the

given constraints of a motor drive to make the same drive applicable to a wider range of

applications. These constraints include voltage and current limitations as well as maximum

permissible power loss. Lower current and voltage requirements translate into cost savings as far

as the drive is concerned.

(b) Back emf: This is the voltage induced in the windings by the rotating magnetic field in the air

gap. Basically, the phase voltage has to work against the back emf in order to enforce a desired

current. For most analytical purposes, specially at higher than base speeds, the fundamental

component of the back emf and that of the phase voltage can be assumed to be approximately the

same in steady state. The small difference is due to the voltage drop across the phase resistance.

(c) Maximum torque vs. speed profile: This profile is frequently used to match a motor drive to

an application. A larger and wider torque vs. speed profile results in the motor drive being

suitable for a wider range of applications.

(d) Power: Amount of work delivered divided by time. This performance index is also referred

to as real power.


58

(e) Volt Ampere: The product of phase voltage and phase current multiplied by the number of

phases. Also referred to as apparent power.

(f) Power factor: Power divided by Volt Ampere. A measure of the level of utilization of the

available apparent power.

(g) Power loss: This index can include copper and core losses, drive losses, windage, friction and

stray losses of a machine. The maximum permissible power loss influences the operational

boundary of a motor drive.

(h) Efficiency: Shaft power divided by input power. This is a very critical index for many

applications.

(i) Torque per unit current: A higher value of this index indicates that more torque is being

produced for a given magnitude of current.

(j) Torque per unit power loss: A higher value of this index indicates that more torque is being

produced for a given net power losses.

(k) Air gap flux linkages: The net flux linkages crossing the air gap. The product of this index

and electrical speed is back emf. Core losses depend on flux linkages, speed and machine

parameters. Flux linkages are also important from the point of view of magnetic saturation.

(l) d-axis current: d-axis current results in flux linkages along the same path as the magnet's flux

linkages. This current must be limited to the maximum value beyond which the magnets get

partially demagnetized. Therefore, dI serves as an index that needs to be monitored and limited.

(m) Base speed: The speed at which the back emf reaches the maximum possible value along the

maximum torque vs. speed envelop is defined here as base speed. The base speed is a function

of motor parameters, maximum voltage available to the phase, and the control strategies used in

the lower and higher than base speed operating ranges.

(n) Maximum speed: The maximum speed attainable by a motor drive. By definition zero torque

is produced at this maximum speed.

(o) Complexity of implementation: This indicator is used to compare different control strategies

from the point of view of implementation requirements.


59

(p) Current and torque ripples: Current ripple is caused mainly by the drive’s current control

mechanism, or lack of control over current. Machine and drive imperfection also result in

current ripple, but analysis of this type of ripple is beyond the scope of this dissertation. The

drive’s current control mechanism usually results in a low percentage of current ripple with

frequencies in the range of 10 to 30 kHz for common applications. Lack of control over current

usually results in higher magnitudes of ripple with frequencies that are proportional to speed.

Current ripples translate into torque ripples directly.

(q) Cost: This is the most fundamental criteria in motor drives as well as in any other

commercial product. A motor drive must be designed such that the overall cost is minimum

while it meets or exceeds the requirements of an application. The overall cost includes cost of

design, material, manufacturing requirements, warranties and service.

5.3 CONTROL STRATEGIES: LOWER THAN BASE SPEED OPERATING RANGE

The most important objective of high performance control strategies is to maintain linear

control over torque. Therefore, qi and di must be coordinated to satisfy the equation given

below for a desired torque, eT ,

)ii)LL(i(P75.0T qdqdqafe −+λ= (5.1)

However, it can be seen from (5.1) that a wide range of qi and di values yield the same torque.

Each of the five control strategies discussed in sections 5.3.1 to 5.3.5 utilize the available degree

of freedom, seen in (5.1), to meet a particular objective. Equation (5.2) shows the general

description of the intended relationship between currents, qi and di , and torque and speed, eT

and rω , respectively, for a given control strategy,


60

ωΓ

ωΛ=

),T(

),T(

i

i

re

re

d

q (5.2)

where Λ and Γ represent the relationship described by (5.1) in combination with the objective

of the specific control strategy. At any given speed the maximum possible torque is limited by

the maximum possible power loss, lmP . Therefore, qi and di must satisfy the following

equation while the system is operating at maximum torque,

])iL()iL[(R

5.1)ii(R5.1P 2

ddaf2

qq2r

c

2ds

2qsslm +λ+ω++= (5.3)

where, qsi and dsi are defined as a function of qi and di in (3.1). In this chapter the maximum

possible torque at any given speed is studied for different control strategies. Chapter 8 addresses

the analytical details of the derivations involved in calculating the maximum torque vs. speed

profiles. Generally, the maximum possible torque is a function of speed, maximum possible loss

and the chosen control strategy, as described below:

)P,(T lmrem ωΥ= (5.4)

where emT is the maximum possible torque, and the function Υ depends on the chosen control

strategy and machine parameters.

The maximum torque under the maximum efficiency, maximum torque per unit current and

zero d-axis current control strategies is only limited by the maximum possible power loss for the

motor. However, each of the unity power factor and constant mutual flux linkages control

strategies impose an absolute maximum possible torque, as described in sections 5.3.4 and 5.3.5,

respectively. Therefore, for the latter two control strategies, the maximum possible torque is the

smaller of the respective absolute maximum torque and the maximum torque resulting from the

power loss limitation.


61

5.3.1 Maximum Efficiency Control Strategy (ME)

qi and di are coordinated to minimize net power loss, lP , at any operating torque and speed.

The net power loss can be described as given below,

])iL()iL[(R

5.1)ii(R5.1P 2

ddaf2

qq2r

c

2ds

2qssl +λ+ω++= (5.5)

where,

ωλ+

ω−

ω

=

0 R

i

i

1 R

LR

L 1

i

ic

raf

d

q

c

rq

c

rd

ds

qs (5.6)

The optimal set of currents, qi and di , that result in minimization of power loss at a given

speed and torque can be found using (5,1), (5.5) and (5.6). This operation can be performed

using numerical methods. Alternatively, an analytical solution, that yields (5.2) for this control

strategy, is given in Chapter 8.

Equation (5.5) is a simplified representation of the sum of copper and core losses. In practice a

more accurate representation of core losses, based on elaborate equations or tables, can be used

to increase the accuracy of the total loss estimation. Other types of losses, such as drive losses,

friction, windage and stray losses, can also be included. More accurate equations for core losses

can be found in [13, 14].

Minimizing lP at zero speed results in minimizing copper losses since core losses are zero at

zero speed. On the other hand, minimizing copper losses is equivalent to minimizing current.

Therefore, the maximum efficiency control strategy results in minimum current for a given

torque at zero speed, which means that the ME and MTPC control strategies result in identical

performance at zero speed.


62

5.3.2 Zero d-axis Current Control Strategy (ZDAC)

The torque angle is defined as the angle between the q-axis current and the d-axis in the rotor

reference frame. This angle is maintained at 90 degrees in the case of the ZDAC control

strategy. The ZDAC control strategy is the most widely utilized control strategy by the industry.

The d-axis current is effectively maintained at zero in this control strategy. The main advantage

of this control strategy is that it simplifies the torque control mechanism by linearizing the

relationship between torque and current. This means that a linear current controller results in

linear control over torque as well. In dc motors the current and magnet fields are always

maintained at an angle of 90 degrees. Therefore, the ZDAC control strategy makes a PMSM

operate in a similar way to the dc motor. This makes the ZDAC control strategy very attractive

for industrial designers who are used to operating dc motor drives. The following relationships

hold for the ZDAC control strategy:

safe iP75.0T λ= (5.7)

where si is the phase current magnitude, and

sq ii = (5.8)

0id = (5.9)

The current, si , for a given torque, eT , can be calculated as,

af

es P75.0

Ti

λ= (5.10)

The air gap flux linkages can be described as,

5.02s

2d

2afm )iL( +λ=λ (5.11)

The ZDAC control strategy is the only control strategy that enforces zero d-axis current. This

is one of the disadvantages of this control strategy as compared to the other four control

strategies. A non-zero d-axis current has the advantage of reducing the flux linkages in the d-


63

axis by countering the magnet flux linkages. This serves to generate additional torque for inset

and interior PMSM, and also reduces the air gap flux linkages. Lower flux linkages result in

lower voltage requirements as well. Therefore, application of the ZDAC control strategy results

in higher air gap flux linkages and higher back emf as compared to other control strategies. The

maximum possible torque under this control strategy is only limited by the maximum possible

power loss.

5.3.3 Maximum Torque per Unit Current Control Strategy (MTPC)

The MTPC control strategy is the most widely studied control strategy by the research

community. Application of this control strategy results in the production of maximum possible

torque at any given current. This control strategy minimizes current for a given torque.

Consequently, copper losses are minimized in the process. The additional constraint imposed on

qi and di for motors with magnetic saliency is:

)LL

i(iiqd

afdd

2q −

λ+= , qd LL ≠ (5.12)

For the types of PMSM that do not exhibit magnetic saliency the MTPC and ZDAC control

strategies are the same. The MTPC control strategy results in maximum utilization of the drive

as far as current if concerned. This is due to the fact that more torque is delivered for unit

current as compared to other techniques. The MTPC and ME control strategies result in

identical current commands at zero speed. The maximum possible torque under this control

strategy is only limited by the maximum possible power loss.

5.3.4 Unity Power Factor Control Strategy (UPF)

Power factor can be defined as given below if the voltage drop across the phase resistance is

ignored,

)EIcos()cos(pf ∠−∠=θ= (5.13)

where I and E are the current and back emf phasors, respectively, and “∠ ” denotes the angle

of the respective phasor. The angle θ can be described as,


64

)i

i(tan)

iLiL

(tand

q1

qq

ddaf1 −− −−

+λ=θ (5.14)

Unity power factor can be achieved by maintaining the following relationship between qi and

di ,

0iiLiL daf2qq

2dd =λ++ (5.15)

This control strategy imposes an absolute maximum possible torque on the system. This

maximum permissible torque is found by inserting qi as a function of di from (5.15) into the

torque equation (5.1), and differentiating torque with respect to di . By differentiating this

equation and equating it to zero the d-axis current, dmI , that yields the maximum possible

torque, emT , is derived. The following equation yields dmI ,

0II dm2dm =γ+β+α (5.16)

where,

dqd L)LL(4 −=α

dafafqd L2)LL(3 λ+λ−=β

2afλ=γ

Inserting dmI into (5.15) yields the q-axis current, qmI , at absolute maximum torque. Inserting

qmI and dmI in (5.1) yields the absolute maximum torque possible under the UPF control

strategy.

5.3.5 Constant Mutual Flux Linkages Control Strategy (CMFL)

By maintaining a constant air gap flux linkages the machine is protected against magnetic

saturation. Also, the maximum speed after which flux weakening becomes necessary is

extended. The most common choice is to maintain the air gap flux linkages at the magnet’s flux

linkages value. In this case equation (5.17) must hold.


65

2qq

2ddaf

2af )iL()iL( ++λ=λ (5.17)

This control strategy imposes an absolute maximum torque on the system. This maximum

permissible torque is found by inserting qi as a function of di from (5.17) into the torque

equation (5.1), and differentiating torque with respect to di . By differentiating this equation and

equating it to zero the d-axis current, dmI , that yields the maximum possible torque, emT , is

derived. The following equation yields dmI in this case,

0III dm2dm

3dm =ϑ+γ+β+α (5.18)

where,

2qd

2d )LL(L4 −=α

)LL2)(LL(L6 qdqddaf −−λ=β

)L4L5(L2 qd2afd −λ=γ

d3af L2λ=ϑ

The smallest real and negative solution of (5.18) is the right choice. qmI can then be calculated

by inserting dmI into (5.17). Inserting dmI and qmI into (5.1) yields the absolute maximum

possible torque under the CMFL control strategy. This maximum torque is usually very high for

PMSM, and requires excessive current.

5.4 CONTROL STRATEGIES: HIGHER THAN BASE SPEED OPERATING RANGE

The fundamental component of phase voltage is constant in the higher than base speed, or flux

weakening, operating range. Voltage restriction, as well as the requirement for torque linearity,

limit the number of possible control strategies in the flux weakening range. The main

possibilities in this range are the six-step voltage (SSV) and constant back emf (CBE) control

strategies. The CBE control strategy limits the back emf to a desired value such that a constant

voltage margin is preserved throughout the flux weakening range. This margin allows the

system to retain instantaneous control over current and torque in the flux weakening range. In


66

this range back emf and instantaneous torque are controlled by coordinating the current phasor’s

magnitude and angle with respect to the rotor. This type of control is only suitable for a limited

spectrum of high performance applications that require high quality control over torque at higher

than base speed. In the SSV control strategy maximum possible voltage is always applied to the

phases, and only the average torque can be controlled. The average torque is controlled by

varying the angle of voltage phasor with respect to the rotor’s magnetic field.

In this section the CBE and SSV control strategies are analyzed. The operational envelope,

current requirements and maximum speed are studied in each case based on the concept of

constant power loss. Current and torque ripples are also studied in each case. The maximum

torque vs. speed envelope for both control strategies is found by limiting the net power loss of

the machine to a desired level. The motor always operates within a safe thermal boundary by

limiting the net power loss to a desired value. The most dominant types of power losses, which

are core and copper losses, are included in this analysis. Analytical derivations of the maximum

speed and maximum current requirement in the flux weakening range are derived for each

control strategy. The fundamental current ripple for the CBE control strategy is the result the

enforcement of instantaneous control over current. The fundamental current ripple for the SSV

control strategy results from the lack of instantaneous control over current. A procedure for

deriving the instantaneous phase current and torque waveforms in steady state under the SSV

control strategy is given for PMSM. This procedure is an application of the analysis presented in

[34].

Section 5.4.1 provides the basics of the CBE control strategy. The details of the SSV control

strategy are presented in section 5.4.2.

5.4.1 Constant Back emf Control Strategy

The CBE control strategy is discussed here. The operational envelope and current requirements

of the system are studied.


67

A. The Basics

In this mode the d and q axis currents, qi and di , respectively, are coordinated to achieve a

desired constant back emf as well as torque linearity. qi and di need to satisfy equations (5.19)

and (5.20) in steady state at a given speed, rω , and for a desired back emf, mE , and a desired

torque, eT .

2qq

2ddaf

2

r

m )iL()iL()E

( ++λ=ω

(5.19)

)ii)LL(i(P75.0T dqqdqafe −+λ= (5.20)

Equation (5.19) ensures that the back emf remains constant at a desired value mE . Equation

(5.20) ensures torque linearity. Note that in the flux weakening range each of the two currents is

a function of both speed and torque as described below,

ωΓ

ωΛ=

),T(

),T(

i

i

re

re

d

q (5.21)

where Λ and Γ represent the relationships as described by (5.19) and (5.20). The

implementation of these equations can be performed in software or using programmable read-

only memory chips utilized as two dimensional look-up tables in hardware [44].

B. Maximum Current in the Flux Weakening Range

The maximum current in a machine in steady state is determined by the designed maximum

permissible power loss. The electrical power losses in a machine are copper and core losses.

Equation (5.22) provides a simplified estimation of the net power loss for a PMSM.

2m

2r

c

2ssl R

5.1IR5.1P λω+= (5.22)


68

where mλ is the air gap flux linkages as described below,

5.02qq

2ddafm ])IL()IL[( ++λ=λ (5.23)

The second term on the right hand side of (5.22) represents the core losses of the motor.

The product of speed and air gap flux linkages is equal to the desired constant back emf in the

flux weakening range. On the other hand the maximum permissible torque at a given speed

occurs when the net power loss is equal to the maximum permissible power loss, lmP . lmP is

assumed to be a constant for the full range of operating speed. Therefore, it can be concluded

from (5.22) that,

c

2m2

smslm RE5.1IR5.1P += (5.24)

where, smI is the value of current along the operational boundary. It is noted that the core losses

are constant in the flux weakening range as the induced emf is kept a constant. This is seen from

the part of (5.24). Therefore, for constant net power loss, the phase current magnitude is also a

constant and this is inferred from (5.24) as well. Equation (5.24) defines the current requirement

in the flux weakening range as a function of maximum permissible power loss and desired back

emf for the CBE control strategy.

C. Operational Boundary

The maximum possible torque at a given speed in the flux weakening range is the torque that

results in maximum possible power loss. The current phasor at which maximum torque is

produced while both back emf and net power loss are at maximum level must satisfy equations

(5.25) and (5.26) as described below,

2qmq

2dmdaf

2

r

m )IL()IL()E

( ++λ=ω

(5.25)


69

cs

2m

s

lm2qm

2dm

2sm RR

ER5.1

PIII −=+= (5.26)

where the current phasor is the resultant of q and d axis currents, qmI and dmI , respectively, at

maximum torque. Note that it is assumed that the rotor current and phase current are

approximately equal as discussed in section 3.7. Inserting qmI and dmI into (5.20) gives the

maximum permissible torque at speed rω . Therefore, the operational envelope can be described

as,

)P,(T lmrem ωΦ= (5.27)

where Φ represents the concurrent solution of (5.25), (5.26) at (5.20) at a given speed.

D. Maximum Speed in the Flux Weakening Range

At maximum speed torque is zero. At this operating point maximum possible current is utilized

to counter the magnet flux linkages, and qI is zero. Therefore, it can be concluded from (5.25)

that the maximum speed is given as,

5.0

sc

2m

s

lmdaf

m

smdaf

mrm

)RR

ER5.1

P(L

EIL

E

−−λ

=−λ

=ω (5.28)

where, rmω is the maximum speed, and smI is given in (5.26).

5.4.2. Six Step Voltage Control Strategy

An alternative to the CBE control strategy is the six step voltage (SSV) control strategy. In this

control strategy full bus voltage is applied to the motor. In this case the phase voltage is quasi-

sinusoidal with six steps such as the one shown later in Fig. 5.1. Average torque is controlled by


70

varying the angle of the voltage phasor with respect to the rotor field. The magnitude of the

voltage phasor is determined by the dc bus voltage.

A. Fundamental Analysis

The average torque produced in SSV mode can be calculated from the fundamental component

of the six step input voltage. This is based on the fact that only the fundamental components of

variables contribute to the average torque. The peak of the fundamental component of voltage,

mV , available to the motor phase for a 3-phase star-connected motor with a full bridge power

stage is described as,

dcm V)2

(Vπ

= (5.29)

where, dcV is the bus voltage. The fundamental components of q-axis and d-axis voltages in the

rotor reference frame, qsV and dsV , can be expressed as,

)sin(VV mqs α= (5.30)

)cos(VV mds α= (5.31)

where α is the angle of the voltage phasor with reference to the rotor's d-axis. The voltage drop

across the stator resistance is negligible since this voltage drop is small relative to phase voltage

in the flux weakening range. Therefore, qV and dV in steady state can be described as,

)IL( V ddafrqs +λω= (5.32)

qqrds ILV ω−= (5.33)

The axes currents can be derived from (5.30)-(5.33) and inserted in (5.20) resulting in the

following relationship between torque and the angle of voltage phasor:

)]2sin(L2

V)LL()cos()[

LV

(P75.0Tqr

mqdaf

dr

me α

ω

−+αλ

ω−

= (5.34)


71

Equation (5.34) can be used to calculate α as a two dimensional function of rω and desired

torque, eT , as given below,

),T( re ωΚ=α (5.35)

where the function Κ is obtained from (5.34). The implementation of such an equation can be

performed in software or in hardware using a programmable memory chip [44].

As far as the fundamental components of variables are concerned the CBE and SSV control

strategies can be evaluated using the same basic equations such as (5.19) and (5.20). The only

difference is that the peak voltage available to the phase is different in each case. For the CBE

control strategy the peak voltage is mE , and for the SSV control strategy the peak voltage is mV

as given by (5.29). Therefore, the same arguments made in the last section about maximum

speed and current for the CBE control strategy are equivalently valid for the fundamental

analysis of the SSV control strategy.

B. Steady State Current in SSV Mode

The steady state performance has been calculated so far using the fundamental components of

the input voltages only. The steady state performance for the actual input voltages including

harmonics is necessary to select the rating of the converter-inverter switches and computation of

losses, and for derating of the PMSM. The steady state then is calculated by either using steady

state harmonic equivalent circuits and summing the responses or directly by matching the

boundary conditions. The harmonic equivalent circuit approach has the conceptual advantage of

simplicity but carries the disadvantage of accuracy being limited by the number of harmonics

considered in the input voltages. The direct steady state evaluation overcomes this disadvantage

but is limited due to the requirement of a computer for solution. This method is derived and

discussed here.

The direct method exploits the periodicity of input voltages and currents in steady state. As

these variables are periodic over a given interval their boundaries are matched to extract an

elegant solution [46]. In the SSV mode the instantaneous phase voltages, av , bv and cv , are


72

balanced six step voltages in steady state. The instantaneous q and d axis voltages, qv and dv ,

respectively, are periodic with a period of 3/π electrical radians in steady state. Accordingly,

the q and d axis currents are also periodic with the same period. qi and di are both continuous

variables. However, qv and dv are not necessarily continuous variables. The complete

procedure is described in section 5.7. Fig. 5.1 shows the normalized variables qsv , dsv , qi , di ,

eT , ai , and av , respectively, for the PMSM in SSV mode with parameters given in Appendix I.

The subscript “n” denotes that the respective variable is normalized using rated values. In this

simulation the voltage phasor angle α is chosen to be 116 o and speed is 1635 RPM with a bus

voltage of 65 V.

C. Operational Boundary for the SSV Control Strategy

The maximum torque at a given speed under the SSV control strategy can be found by

observing the fundamental components of phase voltages and currents. The procedure for

finding the operational boundary for the SSV control strategy is similar to the procedure used in

section 5.4.1 for the CBE control strategy. The only difference is that the maximum desired back

emf, mE , is replaced by mV as defined in (5.29). Therefore, the fundamental value of current

under the SSV control strategy is,

5.0

sc

2m

s

lmsm )

RRV

R5.1P(I −= (5.36)

The maximum permissible torque at a given speed can be found by solving (5.25) and (5.36) to

find the q and d axis currents, and then inserting these currents in (5.20). It is noted that the

actual net power loss is slightly higher than that calculated based on the fundamental component

of current due to the additional losses caused by current harmonics.


73

p.u. ,vqsn

p.u. ,vdsn

p.u. ,iqn

p.u. ,idn

p.u. ,Ten

p.u. ,ian

p.u. ,van

degrees electrical angle, rotor

Fig. 5.1. Normalized motor variables in steady state for a six step voltage input vs. rotor angle in

electrical degrees at 1635 RPM, dc bus voltage of 65 Volts and voltage phasor angle of 116 degrees.

D. Maximum speed, fundamental current and peak current

The maximum speed for the SSV control strategy can be calculated using (5.28) by

substituting mV for mE . The fundamental current for the SSV control strategy can be


74

calculated using (5.36) by substituting mV for mE . The peak current can be calculated using

the procedure outlined later in section 5.7.

5.5 COMPARISON OF CONTROL STRATEGIES BASED ON THE CPL CONCEPT

Any of the five control strategies discussed in section 5.3 can be used in the lower than base

speed operating range. In the higher than base speed operating range either the CBE or SSV

control strategy can be implemented. Therefore, ten different combinations are possible to

cover the full range of operating speed. Each combination results in a unique operational

envelope. The performance of the system along each envelope is studied in this section. The

procedure for comparing the performance of the system under each combination is presented

here using the parameters of the PMSM motor drive prototype described in Appendix I. The

same procedure can be applied to any motor drive.

The five control strategies for the lower than base speed operating range each result in unique

performances. However, the SSV and CBE control strategies are very similar from a

fundamental point of view as discussed in section 5.4.

Section 5.5.1 compares the performance of a motor drive for the full range of speed with the

emphasis being placed on the lower than base speed control strategies. The back emf is limited

to 1.1 p.u. for the flux weakening range. A comparison between the CBE and SSV control

strategies is presented in section 5.5.2.

5.5.1 Lower than Base Speed Operating Range

Several key performance indices are evaluated here vs. speed for each of the five control

strategies discussed in section 5.3. Maximum torque, current, power, torque per current, back

emf and power factor are chosen for this purpose. The maximum torque vs. speed envelope

determines if a motor drive can meet the torque requirements of a particular application. The

current vs. speed envelope determines part of the requirements imposed on the drive. Higher

current requirements translate into a more expensive power stage in the drive. The torque per

unit current index is one of the most common performance indices used by researchers.


75

Therefore, the torque per unit current index vs. speed is also studied here for each control

strategy. Power vs. speed shows the maximum possible real power at any given speed for a

particular control strategy. The power factor vs. speed figure shows how well a particular

control strategy utilizes the apparent power.

Figure 5.2 shows normalized maximum torque, current, power and power loss vs. speed for

each of the five control strategies discussed earlier. The net power loss is maintained at rated

level. The CBE control strategy, with mE of 1.1 p.u., is chosen for the flux weakening range.

Figure 5.3 shows torque per current, back emf, power factor and net loss vs. speed trajectories at

rated power loss. The motor drive parameters are given in Appendix I. All variables have been

normalized using rated torque, current and speed. All operational envelopes are calculated using

assumption (3.3) as discussed in Chapter 3. The slight deviation of the power loss vs. speed

figures from 1 p.u. is due to this assumption. The performances of the five control strategies are

compared below based on Figures 5.2 and 5.3.

A. Torque vs. Speed Envelope

The ME control strategy provides more torque at any speed than any other control strategy.

The MTPC control strategy provides only slightly less torque. However, the maximum torque

for the MTPC control strategy drops at a faster rate as speed increases. The ZDAC control

strategy provides the least torque at any given speed. This is mostly due to the fact that the

ZDAC control strategy does not utilize the machine’s reluctance torque in the case of this

example. The maximum torque vs. speed envelope for the UPF and CMFL control strategies

fall in between the ZDAC and ME control strategies. The UPF control strategy produces slightly

more torque than the CMFL control strategy at all speeds. Note that while the ME control

strategy generates more torque than the MTPC control strategy it also requires more current at

any given speed.


76

Fig. 5.2. Maximum torque, current, power and net loss vs. speed at rated power loss for lower

than base speed control strategies, and 1.1Em = p.u. for the flux weakening range.


77

Fig. 5.3. Torque per current, back emf, power factor and net loss vs. speed at rated power loss for

lower than base speed control strategies, and 1.1Em = p.u. for the flux weakening range.


78

The maximum torque for the ZDAC control strategy actually rises between the speeds of 0.7 and

0.9 p.u. speed. This is due to the introduction of non-zero d-axis current in the flux weakening

range, which produces reluctance torque.

B. Current vs. Speed

All control strategies require the same current at zero speed under the constant power loss

criteria. This is due to the fact that core losses are zero at zero speed. Therefore, constant net

power loss implies constant copper losses at zero speed. And, constant copper losses result in

identical current magnitude for all control strategies at zero speed. As speed increases the

current requirements of the five control strategies diverge significantly. The CMFL control

strategy has the highest current requirement. The rate of drop of current increases successively

for each of the UPF, ME, MTPC and ZDAC control strategies.

C. Power vs. Speed

The ME control strategy produces more power at any given speed than any other control

strategy. The power levels drops successively for each of the MTPC, UPF, CMFL and ZDAC

controls strategies.

D. Torque per Current vs. Speed

The ME and MTPC control strategies produce almost the same torque per current vs. speed

envelope. The ZDAC and CMFL control strategies result in roughly 1 p.u. torque per current for

the full range of speed. The toque per current envelope for the UPF control strategy is only

slightly higher than that of the CMFL control strategy.

E. Back EMF vs. Speed

The ZDAC control strategy results in the highest back emf among the five control strategies.

This, in turn, significantly limits the speed range before flux weakening for the ZDAC control

strategy. The UPF and CMFL control strategies result in the least back emf among the five

control strategies, and lower back emf requirements leads to increased speed range before flux


79

weakening for these two control strategies. The ME and MTPC control strategies also have

similar back emf requirements, and their back emf requirements are significantly higher than that

of UPF and CMFL control strategies.

F. Air gap Flux Linkages vs. Speed

Higher back emf at a given speed indicates higher mutual flux linkages as well. Therefore,

mutual flux linkages requirements can be studied using back emf vs. speed figures. The ZDAC

control strategy requires by far the highest air gap flux linkages of all control strategies at any

speed. This may raise concerns regarding saturation of the core for some machines. The ME

and MTPC control strategies both require roughly the same air gap flux linkages. The CMFL

and UPF control strategies require the least mutual flux linkages among all control strategies.

The CMFL and UPF control strategies both require roughly the same back emf. Therefore, the

flux linkages of the UPF control strategy is roughly 1 p.u., i.e. almost the same as that of the

CMFL control strategy.

G. Power Factor vs. Speed

The UPF control strategy results in the highest possible power factor of 1 for the full range of

speed. The CMFL control strategy results in a nearly unity power factor as well. The ZDAC

control strategy results in the worst power factor of all five control strategies. The power factor

is roughly 0.65 on the average in this case. The ME and MTPC control strategies both result in

reasonable power factors ranging from 0.85 at lower speeds to 0.95 at higher speeds. Power

factor increases for both of these control strategies as speed increases.

H. Speed Range Before Flux Weakening

The CMFL control strategy results in the widest speed range before flux weakening among the

five control strategies. The UPF control strategy stands with a slightly lower speed range. The

ZDAC control strategy yields the narrowest speed range. This is mainly due to the relatively

large air gap flux linkages for this control strategy. Note that the ZDAC control strategy is the

only control strategy where the magnet flux linkages is never opposed by a countering field in

the rotor’s d-axis. The speed ranges for the ME and MTPC control strategies fall in between that


80

of the ZDAC and UPF control strategies. The ME control strategy results in a slightly higher

speed range than the MTPC control strategy.

I. Base Speed

The speed at which the back emf reaches the maximum possible value along the maximum

torque vs. speed envelop is defined here as base speed. The base speed is a function of

maximum permissible power loss, maximum voltage available to the phase, and the choice of

control strategies. The specific objective of a control strategy in the lower than base speed

operating range cannot be met beyond the base speed along the operational envelope of the

system. It is seen that the CMFL control strategy provides the largest base speed among the five

control strategies. The base speed reduces successively for each of the UPF, MTPC, ME and

ZDAC control strategies.

J. Complexity of Implementation

The ZADC control strategy is the simplest as far as implementation is concerned. dI is simply

maintained at zero which makes the torque proportional to the phase current. The MTPC,

CMFL and UPF control strategies all require the implementation of separate functions for each

of the d and q axis currents. These currents are functions of torque only. Therefore,

implementation of the MTPC, CMFL and UPF control strategies involves the same level of

complexity. However, in the case of the ME control strategy the currents, qi and di , are

functions of both torque and speed. The necessary equations, in all cases, can be implemented

on-line, or by implementing look-up tables [44].

5.5.2 Higher than Base Speed Operating Range

In this section the two control strategies for the flux weakening range are compared in terms of

maximum torque vs. speed envelope, current requirements, base speed, maximum speed, current

and torque ripples, large signal torque response, and implementation complexity. The SSV

control strategy is shown to provide a significantly larger maximum torque vs. speed envelope

for operation under constant power loss. A larger operational envelope translates into a wider


81

speed range as well. The fundamental component of current in the case of SSV control strategy

is shown to be lower in magnitude than that of the CBE control strategy. It is also shown that the

application of the SSV control strategy results in a wider speed range below base speed before

flux weakening needs to be initiated. It is shown that the magnitude of the six-step current and

torque ripples are relatively low. The relatively low magnitude of ripple can be attributed to the

fact that in PMSM the phase self inductance limits the current ripple. It is shown that the SSV

control strategy is simpler to implement as it requires only position information contrary to the

CBE control strategy requiring in addition information on two phase currents. This increases the

sensor requirements for CBE control strategy compared to the SSV strategy.

A. Operational Boundary

Fig. 5.4 shows the operational boundary for the CBE control strategy with mE = 0.8 p.u., bus

voltage of 1.72 p.u., and rated total power loss for the motor drive as described in Appendix I.

The operational boundary for the SSV control strategy is also shown under the same conditions.

The voltage drop across the stator resistance is assumed to be negligible in this simulation. But,

stator resistance is taken into account in calculating power losses. It is seen that the operational

boundary for the SSV control strategy is significantly larger than that of the CBE control

strategy.

B. Base Speed

The starting point for the operational boundary for each control strategy in Fig. 5.4 is the base

speed for that control strategy. It is seen that the SSV control strategy provides a higher base

speed than that provided by the CBE control strategy. Therefore, the SSV control strategy

allows for a wider range of operation before flux weakening is initiated.

C. Maximum Speed in the Flux Weakening Range

As the maximum available voltage to the phase increases the maximum possible speed also

increases. The maximum speeds for the SSV and CBE control strategies are 4877 and 6374

RPM, respectively, as calculated from (5.28) for the example of Fig. 5.4.


82

p.u. ,Ten

p.u. ,Isn

p.u. ,Pln

SSV control strategy


100x torqueaverageripple torque

%

ripple torque.u.p

enω

p.u. 8.0Em =


CBE control strategy

Fig. 5.4. Maximum torque, peak current, net power loss, torque ripple as a percentage of average

torque, and peak to peak torque ripple for SSV (solid lines) and CBE (dashed lines, mE =0.8

p.u.) control strategies.


83

D. Maximum Current Requirements

It is shown in section 5.4.1 that the fundamental component of current is constant along the

operational boundary in the flux weakening range for both control strategies. The fundamental

voltage available to the phase is always higher in the case of the SSV control strategy.

Therefore, it can be concluded from (5.36) that the fundamental component of current is always

lower in the case of the SSV control strategy. Maintaining the back emf at a lower value than

mV requires a significant d-axis current to oppose the magnet flux linkages. This results in a

higher stator current for the CBE control strategy. This additional current does not contribute

proportionally to torque development.

The drive must be able to provide the peak current for any control strategy. The peak current

for the SSV control strategy is different from the fundamental component of current. This is due

to presence of significant current ripple. Fig. 5.4 shows the peak current along the operational

boundary for the SSV control strategy. It is seen that in the worst case the peak value is about

10% higher than the fundamental value of 1.1 p.u. This brings the peak current requirement of

the SSV control strategy to approximately 1.2 p.u. for the given example. A realistic comparison

on the basis of current requirements must compare the peak current in the case of SSV control

strategy to the fundamental current for the CBE control strategy. Fig. 5.4 shows that the

fundamental component of current under the CBE control strategy is slightly over 1.2 p.u. for a

desired back emf of 0.8 p.u. It is seen that in this example the current requirements of both

control strategies are in the same order. However, if the desired back emf can be increased

above 0.8 p.u., then the current in the CBE control strategy drops below the peak current in the

SSV control strategy.

E. Torque Ripple

The torque ripple, resulting from the application of the CBE control strategy, is caused by the

PWM or hysteresis current control system. The impact of this torque ripple on system

performance is negligible as compared to the torque ripple inherent to the SSV control strategy.


84

Fig. 5.4 shows the peak to peak torque pulsation as a percentage of average torque along the

maximum torque vs. speed envelope for the SSV control strategy. It is seen that the peak to peak

torque pulsation remains within 7 to 20 percent of the average torque along the operational

boundary. But, it is noted that in absolute normalized unit, the torque pulsation is less than 0.1

p.u. at all speeds.

F. Large Signal Torque Response

The CBE control strategy provides a faster torque response as compared to the SSV control

strategy. This is due to the availability of a voltage margin in the CBE control strategy.

Simulation results for the motor drive described in Appendix I show that the CBE control

strategy with a 10 volts margin at a DC bus voltage of 57 V at 1500 RPM results in a time

response of 8 ms for a –1 to 1 N.m. step torque command. The SSV control strategy results in a

20 ms time response for the same transition.

G. Complexity of Implementation

The SSV control strategy is easier to implement than the CBE control strategy. Only one

lookup table is required to provide the angle of voltage phasor as a function of torque command

and speed. However, in the case of the CBE control strategy, one look-up table is required for

each of q and d-axis current commands as a function of torque command and speed. Also, the

CBE control strategy requires current feedbacks in addition to rotor position feedback.

However, the SSV control strategy only requires rotor position feedback. The switching losses

of the drive under the SSV control strategy are much less than that resulting from the CBE

control strategy. This is due to the fact that the switching frequency for the SSV control strategy

is much lower than that of the CBE control strategy.

5.6 PERFORMANCE COMPARISON INSIDE THE OPERATIONAL ENVELOPE

In the last section the performances of different control strategies along their respective

operational envelope are studied. However, for a given control strategy, and at a given speed,

the torque command may be anywhere between zero and maximum possible torque at that speed.

In this section the performance of different control strategies are compared as a function of


85

torque. Current, air gap flux linkages, power factor and d-axis current vs. torque are studied for

the ZDAC, MTPC, UPF and CMFL control strategies. This comparison provides insight into the

performances of different control strategies inside their respective operational boundaries. The

ME control strategy is omitted from this study as its performance depends on speed as well as

torque. Note that for very low speeds the performance of the ME and MTPC control strategies

are very similar. Figure 5.5 shows normalized current, air gap flux linkages, power factor and

d-axis current vs. torque for the four control strategies named above for the motor drive with

parameters given in Appendix I. The performance evaluation and comparison is given below in

several categories:

5.6.1 Current vs. Torque

The MTPC control strategy results in lower current requirements at any given torque compared

to the other three control strategies. The ZDAC and CMFL control strategies have similar

current requirements for the full range of torque. The UPF control strategy also has similar

current requirements to the ZDAC control strategy up to a torque value of 0.6 p.u. However, the

current requirements for this control strategy increase significantly for torque levels higher than

0.6 p.u.

5.6.2 Air Gap Flux Linkages vs. Torque

The flux linkages requirements for the UPF control strategy is roughly 1 p.u. up to a torque

level of 1.2 p.u. However, the flux linkages requirement of the UPF control strategy actually

drop for the higher than 1.2 p.u. torque range. The CMFL control strategy enforces unity mutual

flux linkages requirement by definition. The ZDAC control strategy has the highest flux

linkages at any given torque. The MTPC control strategy’s mutual flux linkages is lower than

that of the ZDAC control strategy, but higher than the mutual flux linkages of the UPF and

CMFL control strategies. The high flux linkages requirements for the ZDAC and MTPC control

strategies may result in core saturation at higher torque levels.

5.6.3 Power Factor vs. Torque

The CMFL and UPF control strategies yield nearly unity power factor for the full range of

torque. The power factor for the MTPC control strategy consistently falls as torque increases.


86

Fig. 5.5. Current, air gap flux linkages, power factor and d-axis current vs. torque for different

control strategies.


87

However, its power factor remains above the reasonable value of 0.8 for the full range of torque.

The power factor for the ZDAC control strategy falls significantly as torque command increases.

5.6.4 D-axis Current vs. Torque

The ZDAC control strategy enforces zero d-axis current on the motor. All the other control

strategies require an increasing magnitude of negative d-axis current as torque increases. The d-

axis current requirement of the UPF and CMFL control strategies are similar and relatively

significant. The MTPC control strategy requires reasonable d-axis current for the full range of

torque. High levels of d-axis current can demagnetize the magnets.

5.6.5 Torque Range

The UPF control strategy has the smallest torque range among the four control strategies. This

is due to the fact that beyond a certain torque the power factor cannot be maintained at 1 as

discussed in section 5.3.4. The other control strategies do not have any absolute torque limit

within the shown range. However, both air gap flux and d-axis current limitations can impose

torque limits on these control strategies depending on the structure of the motor. The air gap flux

linkages is limited primarily by magnetic saturation. Therefore, application of the ZDAC control

strategy may result in a torque limit because of its relatively high air gap flux linkages

requirements. Similarly, the maximum possible d-axis current imposes torque limitations on the

UPF and CMFL control strategies due to their relatively high d-axis current requirements.

5.7 DIRECT STEADY STATE EVALUATION IN SSV MODE

In this section the boundary matching technique is utilized in finding the instantaneous q and d

axis current waveforms in steady state for a PMSM operating in the SSV mode.

Assuming that the angle of the voltage phasor with reference to the rotor’s d-axis is α then the

q and d-axis voltages can be described in rotor reference frame as,

)3/tcos(V32

)t(v rdcqs α−π+ω= 3/t0 r π<ω≤ (5.37)


88

)3/tsin(V32

)t(v rdcds α−π+ω= 3/t0 r π<ω≤ (5.38)

The same pattern repeats in the 3/2t3 r π<ω≤π period as well as other periods. Note that α is

the independent control variable that can be used to control average torque. The fundamental

components of (5.37) and (5.38) are the same as the values given by (5.30) and (5.31),

respectively, for a given α . The periodic voltage inputs, as described by (5.37) and (5.38), result

in periodic q and d axis currents. The periodicity and continuity of the currents are exploited

here to derive the waveforms of qi and di for one period of 3/π radians. The PMSM equations

in rotor reference frame are written in state variable form as,

eBUBXAX 111111 ++=& (5.39)

where

tdq1 ]ii[X = , 1

11 PQA −= , 1

1 QB −= , tdsqs1 ]vv[u = ,

tafr ]0 [e λω−= ,

=

d

qL00L

Q ,

−ωω−−

=sqr

drs1 RL

LRP

Equations (5.37) and (5.38) can be described in differential format as given below,

2ds

qs

r

r2

.

ds.

qs.

SXv

v

0

0X

v

v=

ω

ω−==

(5.40)

where the dot on top of the variables denotes differentiation with respect to time. Considering

only the part without rotor flux induced emf in the q axis, equations (5.39) and (5.40) are

combined to yield,

AXX.

= , [ ] t21 XXX = ,

=

S0BA

A 11 (5.41)

The solution of (5.41) is written as,


89

)0(Xe)t(X At= r3

t0ωπ

<≤ (5.42)

The initial condition vector is given below,

tdsqsdq )]0( v)0( v)0(i )0(i[)0(X = (5.43)

where )0(X is to be evaluated to compute X(t). X(0) is found using the fact that the state vector

has periodic symmetry and hence,

)0(XS)3

(X 1r

=ωπ

(5.44)

where 1S is evaluated later. The boundary condition for the currents are,

)0(X)3

(X 1r

1 =ωπ

(5.45)

The boundary matching condition for the voltage vector is obtained by expanding (5.37) and

(5.38).

)0(v23

)0(v21

3sin)

3sin(

3cos)

3cos(V

32

)33

cos(V32

)3

(v

dsqs

dc

dcr

qs

−=

π

α−π

−π

α−π

=

π+α−

π=

ωπ

(5.46)

Similarly,

)0(v21

)0(v23

)3

(v dsqsr

ds +=ωπ

(5.47)

Hence

)0(XS)0(X

21

23

23

21

3X 222

r2 =

−

=

ωπ (5.48)


90

Substituting equation (5.45) and (5.48) into (5.42),

)0(Xe)0(XS00I

)0(XS3

X r3A

21

r

ωπ

=

==

ωπ (5.49)

where I is a 2x2 identity matrix. Hence,

0)0(WX)0(X]eS[ r3A

1 ==−

ωπ

(5.50)

where,

=−=

ωπ

43

213A

1 WWWW

]eS[W r (5.51)

where 1W , 2W , 3W and 4W are 22 × matrices. Expanding the upper row in (5.50), gives the

following relationship,

0)0(XW)0(XW 2211 =+ (5.52)

from which the steady state current vector )0(X1 is obtained as,

)0(XWW)0(X 221

11−−= (5.53)

Having evaluated the initial current vector, it could be used to evaluate currents for one full cycle

and the electromagnetic torque. Now using (5.42) the exact current vector as a function of angle

can be derived for one period, starting at t = 0, as,

I

I)0(Xe

0 0 1 00 0 0 1

)t(i

)t(i

d

qAt

d

q

+

=

(5.54)


91

where tdq ]I I[ is the steady state response due to the induced emf by the rotor flux linkages.

As the induced emf is a constant at a given speed, its response in the machine has to also be a

constant in steady state. Hence the d and q axes current derivatives are zero. With that the

steady state currents are,

[ ]t

qd2r

2s

afq2r

qd2r

2s

safrt

dq LLR

L,

LLR

RI,I

ω+

λω−

ω+

λω−= (5.55)

5.8 SIMULATION AND EXPERIMENTAL VERIFICATION

Experimental correlation is provided using a prototype motor drive utilizing an interior PMSM

with parameters given in Appendix I. The experimental PMSM motor drive is set up as a torque

controller for the higher than base speed operating range. An 8 kByte EEPROM is utilized as a

lookup table with torque command and measured speed as its inputs in order to implement the

SSV control strategy. The output of this EEPROM is the angle of voltage phasor. One 8 kByte

EEPROM lookup table is utilized for each of the q and d axis current commands in order to

implement the CBE control strategy. Both q and d-axis current commands are functions of

measured speed and torque commands for the CBE control strategy. Fig. 5.6 shows the

estimated and measured maximum possible torque vs. speed for the CBE and SSV control

strategies. The maximum power loss is set at 30 W in both cases, and the bus voltage is 65 V.

The constant back emf is 33 V for the CBE control strategy leaving a margin of 8 V between the

back emf and the peak of the fundamental voltage available to each phase. The stator voltage

phasor required to maintain the back emf at 33 V is enforced using the SSV control setup by pre-

calculating the required stator voltage phasor on a point by point basis. It is clearly seen that at

any speed the maximum possible torque is much higher for the SSV control strategy. The

measured maximum speeds for the SSV and CBE control strategies are approximately 1900

RPM and 1650 RPM, respectively. These speed limits correlate closely to the calculated values

of 2000 RPM and 1634 RPM, respectively, using (5.28).


92

emT

CBE control strategy

Speed, RPM


.)m.N(

Fig. 5.6. Estimated (dashed lines) and measured (solid lines) maximum torque vs. speed

trajectory for the SSV and CBE control strategies with maximum power loss of 30 W

and .V65Vdc =

Time, 5 ms/div.

av

ai

50 V/div.

2 A/div.

Fig. 5.7. Measured phase voltage and current for the SSV control strategy at 1635 RPM, 116=α

degrees and =dcV 65 V.


93

A ,ia

V ,va

Time, ms Fig. 5.8. Simulated phase voltage and current for the SSV control strategy at 1635 RPM,

116=α degrees and =dcV 65 V.

It is seen in Fig. 5.6 that at lower speeds the measured and estimated maximum torque vs. speed

trajectories diverge. This can be attributed to the fact that at higher torque and current levels the

q-axis inductance reduces due to saturation of the core. Therefore, higher magnitudes of currents

become possible which result in higher torque. Fig. 5.7 shows the measured phase voltage and

current at 1635 RPM, 116=α degrees and =dcV 65 V. It is seen that the actual phase voltage

and current correlate closely to the simulation result provided in Fig. 5.8. The slight differences

in estimated and measured voltages and currents are due to motor drive control imbalance, the

assumption that qL is constant while qL is a function of qI , and the dc bus voltage not being

perfectly steady.

5.9 CONCLUSIONS

A detailed comparison of different control strategies for high performance control of PMSM is

given in this chapter. The equations governing each control strategy are studied. The necessary

foundation for evaluating the performance of each control strategy along the constant power loss


94

operating envelope is laid out. All control strategies are compared based on their performances

along the constant power loss operational envelope as well as their performances inside the

envelope. The procedures provided in this study can be used to choose the control strategy that

optimizes the motor drive system based on the requirements of a particular application.




• Comparison of control strategies along the constant power loss operating envelope

• Comparison of control strategies inside the operating envelope

• Derivation of absolute maximum torque for the unity power factor and constant mutual flux

linkages control strategies

• Derivation of maximum speed, current and torque in the flux weakening range as a function

of maximum power loss and machine parameters.


95

CHAPTER 6

Analysis and Implementation of Concurrent Mutual

Flux Weakening and Torque Control for PMSM

6.1 INTRODUCTION

Concurrent mutual flux linkages and torque control is necessary for the high performance

control of PMSM in the higher than base speed operating range. Implementing such a system

involves the solution of a fourth order polynomial. The parameters of this polynomial depend on

rotor speed, commanded mutual flux linkages and torque as well as machine parameters.

Implementing a fourth order polynomial equation solver is a difficult task, and requires

expensive processors. Two practical implementation strategies are discussed in this chapter.

The first one is based on a lookup table approach. The two-dimensional lookup tables are

prepared offline, and then implemented within a high performance motor drive. It is shown that

the implementation of the lookup tables is relatively straightforward. Dynamic simulation shows

that the lookup table approach results in very high accuracy of control. However,

implementation of the lookup tables requires a significant amount of programmable memory.

An alternative implementation strategy is to replace the lookup tables with polynomials fits.

This approach results in far less memory requirements, but introduces more error and results in a

slower system. A procedure for replacing the polynomials with artificial neural networks is

suggested later in Chapter 7. The maximum possible torque as a function of the desired mutual

flux linkages is studied in this chapter.


• Introduction of two practical implementation schemes, based on lookup tables or

equations, for the concurrent flux linkages weakening and torque control for PMSM

• Comparison of the two implementation schemes

Chapter 6 Analysis and Implementation of Concurrent Flux and Torque Control 96

• Analysis of the maximum possible torque as a function of the desired mutual flux

linkages

The mutual flux linkages weakening strategy and torque control issues are discussed in section

6.2. An analysis of the maximum permissible torque command as a function of the desired

mutual flux linkages is presented in section 6.3. The control system block diagram is discussed

in section 6.4. Two implementation strategies, along with pertinent simulation results, are

presented in section 6.5. Section 6.6 is devoted to the comparison of the implementation

strategies discussed in this chapter. The conclusions are summarized in section 6.7.

6.2 MUTUAL FLUX LINKAGES WEAKENING STRATEGY AND CONTROL

In this section the mutual flux linkages control strategy is discussed, and the basic equations

governing the motor’s mutual flux linkages and torque are presented. These equations are

utilized in enforcing the desired mutual flux linkages and torque commands on a PMSM.

6.2.1 The Mutual Flux Linkages Weakening Strategy

The following equation provides the relationship between the desired back emf, speed and

mutual flux linkages of a PMSM,

mrmE λω= (6.1)

Therefore, the flux linkages command, *mλ , in the flux weakening range is calculated as given

below,

r

m*m

Eω

=λ (6.2)

If the required flux linkages for a given control strategy is less than that defined in (6.2) then flux

weakening is not necessary.

6.2.2 Mutual Flux Linkages and Torque Control

The q and d axis current commands, *qi and *

di , in the flux weakening range must satisfy the

following two equations:


)ii)LL(i(P75.0T *d

*qqd

*qaf

*e −+λ= (6.3)

2*qq

2af

*dd

2*m )iL()iL( +λ+=λ (6.4)

where *mλ and *

eT are the mutual flux linkages and torque commands, respectively. Equation

(6.3) yields,

)i)LL((P75.0

Ti

af*dqd

*e*

qλ+−

= (6.5)

Inserting (6.5) into (6.4) yields,

2

af*dqd

*eq2

af*dd

2*m )

)i)LL((P75.0

TL()iL(

λ+−+λ+=λ (6.6)

Equation (6.6) can be used to compute *di from the command pair *

m*e ,T λ . *

qi can then be

computed by inserting *di into (6.5). It is evident that solving equation (6.6) for *

di involves the

solution of a fourth order polynomial of *di . The coefficients of this polynomial depend on the

variables *eT and *

mλ . The maximum possible torque for a given mutual flux linkages command

is studied next.

6.3 MAXIMUM TORQUE FOR A GIVEN MUTUAL FLUX LINKAGES

The mutual flux linkages weakening system described in section 6.2 imposes a limit on the

commanded torque. This maximum permissible torque command, *emT , decreases nonlinearly as

the mutual flux linkages command decreases. The following two equations provide the

maximum permissible torque command for a desired mutual flux linkages command value,

x0xx2 afp2*

mpaf2p λ≤=λ−γλ− (6.7)

]x.[]x[L

P75.0T afp

2

12p

2*m

q

*em γλ+−−λα= (6.8)


d

dq

L

LL −=α (6.9)

dq

q

LL

L

−=γ (6.10)

In brief, (6.7) provides the intermediate variable px , which is then used in (6.8) to yield the

maximum permissible torque command for a given mutual flux linkages. The derivation of the

procedure described above is given in Chapter 8. Fig. 6.1 shows such a relationship for a PMSM

with parameters given below: dnL =.434 p.u., qnL =.699 p.u. and snR =.1729 p.u. The subscript

n denotes that the respective variable is normalized. The base values are bV = 97.138 V, bI =

12 A, bL = 0.0129 H and bω = 628.6 rad/s. It can be seen that the )(T *mn

*emn λ relationship is

nearly linear for this example. This means that a first or second order polynomial can be used to

implement this relationship in practice. In general, the function )(T *mn

*emn λ can be computed off

line for the appropriate range of *mnλ , and subsequently incorporated into the system as a simple

lookup table.

6.4 CONTROLLER BLOCK DIAGRAM

The controller block diagram for the flux weakening range is shown in Fig. 6.2. *qI and *

dI are

each functions of the mutual flux linkages and torque commands. The mutual flux linkages

command is calculated as discussed in section 6.2.

p.u.,T*emn

p.u.,*mnλ

Fig. 6.1. Maximum permissible torque command as a function of mutual flux linkages command.


*eT Lookup tables

orequations

r

mE

ω *mλ

rω

*qsiConversion

matrix *dsi

*qi

rω*di

Fig. 6.2. Controller block diagram for the flux weakening range.

The torque command and measured speed are the inputs to this system. The torque command,

*eT , may be an independent variable, or it may be the output of a proportional-integrator

controller acting on the difference between the commanded and measured speeds. The mutual

flux linkages command is calculated as discussed in section 6.2. Either lookup tables or

equations can be used to calculate the d and q axis current commands as a function of the mutual

flux linkages and torque commands. The stator current commands, *dsi and *

qsi , are calculated as

a function of *qi , *

di and rω as shown in the conversion matrix (6.11). This conversion matrix

increases the control accuracy by taking core losses into account. Implementation of (6.11) is

highly recommended at higher speeds. The stator current commands are enforced on the motor

using an appropriate current controller.

ωλ+

ω−

ω

=

0

Ri

i

1R

L

R

L1

i

ic

raf

*d

*q

c

rq

c

rd

*ds

*qs

(6.11)

The controller block diagram for the lower than base speed operating range is shown in Fig.

6.3.

*eT Lookup tables

orequationsrω

*qsiConversion

matrix *dsi

*qi

rω*di

Fig. 6.3. Controller block diagram for the lower than base speed operating range.


The lower than base speed operating range is initiated only if,

r

mm

E

ω<λ (6.12)

where mλ is the required flux linkages. *qi and *

di are each functions of the torque command

and speed in the most general case. Any of the control strategies discussed in section 3 of

Chapter 5 can be implemented as lookup tables or equations in this case. *qsi and *

dsi are

calculated as a function of *qi , *

di and rω as shown in (6.11).

6.5 IMPLEMENTATION STRATEGIES

Two alternative implementation strategies for the concurrent control of mutual flux linkages

and torque are presented in this section. Implementation of either of the two is simpler than the

on-line solution of a fourth order equation described in section 6.2.

6.5.1 Lookup Table Approach

In this section the lookup table implementation strategy for the on-line computation of *q

*d i,i

from *m

*e ,T λ is discussed. All implementation strategies follow the following general format,

Ω=*di ),T( *

m*

e λ (6.13)

Λ=i*q ),T( *m

*e λ (6.14)

where Ω and Λ represent the relationships described in (6.3) and (6.4). The lookup table

approach is illustrated here through an example using the parameters of a PMSM described in

section 6.3.

Equations (6.13) and (6.14) can be realized using separate two-dimensional lookup tables.

These lookup tables are generated off line by numerically solving equations (6.3) and (6.4). The

two independent axes of each table are assigned to *eT and *

mλ , respectively. The respective


values of *di or *

qi is read inside each table. Note that the tables need only provide the data for

positive torque commands. For negative torque commands the q-axis current command from the

table must be applied with negative sign. *mλ is limited to values between 0 and base value

since the only interest is in weakening the mutual flux linkages. The numerical solution to

equations (6.3) and (6.4) for the motor parameters given in section 6.3, and for the following

range of normalized mutual flux linkages command,

12.0 *mn ≤λ≤ , p.u. (6.15)

yields the operating ranges for the other three variables as

44.2T0 *en ≤≤ , p.u. (6.16)

3.3i0 *dn ≤≤ , p.u. (6.17)

3.3i0 *qn ≤≤ , p.u. (6.18)

Figures 6.4 and 6.5 show the two-dimensional tables for this system. In practice the variables

are digitized under the following constraints in order to reduce storage requirements,

:T*en 9 bits (6.19)

:*mnλ 7 bits (6.20)

*dni and *

qni : 8 bits each (6.21)

Once the torque command reaches its maximum permissible value, the current commands are

frozen. The fact that each graph has two distinct areas is due to this restriction. In each graph,

the border between these two surfaces appears to be a straight line, which can be attributed to the

fact that the )(T *mn

*emn λ relationship is nearly linear for this example. The )(T *

mn*emn λ

relationship, shown in Fig. 6.1, suggests that for the given parameters, the relationship can be

adequately approximated with a first or second order polynomial. Thus, any of the following

two polynomials can be used for the on line calculation of *emnT from *

mnλ in this example. The

polynomials are both derived using the least squares fit method.


p.u.,*mnλ

p.u.,T*en

p.u.,i*qn

Fig. 6.4. *qni as a function of normalized mutual flux linkages and torque commands.

p.u.,*mnλ

p.u.,T*en

p.u.,i*dn

Fig. 6.5. *dni as a function of normalized mutual flux linkages and torque commands.

0059.23.221.0T *mn

2*mn

*emn +λ+λ= (6.22)

or simply,

*mn

*emn 44.2T λ= (6.23)

Utilization of these tables results in accurate enforcement of the commanded torque and the

commanded mutual flux linkages. Fig. 6.6 shows simulation results for a speed control system

that utilizes the two tables shown above for a speed controlled operation with +3 p.u. and -3 p.u.


speed commands. The mutual flux linkages is maintained at rated value until the phase voltage

requirement reaches 0.53 p.u. at which point flux weakening is initiated. The stator resistance is

taken into account in the simulation, and this is why the mutual flux weakening is initiated at a

speed of 0.53 p.u. instead of 1 p.u. The torque command is the difference between the

commanded and actual speeds multiplied by a factor of 100. The dotted lines represent

commanded values and solid lines represent actual values. The PWM is operating at 20 kHz and

the current error is amplified by a factor of 200. Due to the limited bandwidth of the current

controller, the ability of the system to enforce the desired currents decreases as speed increases.

As a consequence the mutual flux linkages error increases with speed. It is seen from Fig. 6.6

that the mutual flux linkages and torque commands are enforced accurately. Figure 6 in Chapter

7 is similar to Fig. 6.6, but it also provides the phase voltage and actual and commanded q and d-

axis currents. Note that memory chips are required to store the relatively large look-up tables.

Each table has 162 entries, and each entry is digitized using 8 bits. Therefore, each table

requires 64 kilobytes of storage capacity. This can be considered as a tradeoff for achieving

accurate control over the machine’s mutual flux linkages and torque.

6.5.2 Two-Dimensional Polynomials Approach

An alternative to the lookup table approach is to approximate the tables using two-dimensional

polynomials. In this case equations (6.13) and (6.14) are implemented as,

)(a+...T).(ai *mn0

r*en

*mnr

*dn λ+λ= (6.24)

)(b+...T).(bi *mn0

r*en

*mnr

*qn λ+λ= (6.25)

where,

i0s*

mnsi*mni x+....x)(a +λ=λ (6.26)

i0s*

mnsi*mni y+....y)(b +λ=λ (6.27)

and r is the order of the polynomials that are used to compute *dni and *

qni using equations (6.24)

and (6.25), and s is the order of the polynomials that are used to compute the coefficients of

(6.26) and (6.27).


Time, s

p.u.,*mnλ

p.u.,mnλ

p.u.,T*en

p.u.,T*en

p.u.,*rnω

p.u.,rnω

Fig. 6.6. Normalized speed, torque, mutual flux linkages for a ± 3 p.u. step speed command.

The appropriate order for each polynomial must be chosen on a case by case basis. Obviously,

higher order polynomials provide a better fit at the expense of increased implementation

complexity. The unknown coefficients can be identified for a given PMSM using least squares

fits. The two-dimensional polynomial approach does not have any significant memory storage

requirements. However, the advantage of not having to store large tables comes at a tradeoff for

mutual flux linkages and torque enforcement accuracy. Fig. 6.7 shows the achieved and

commanded mutual flux linkages for various mutual flux linkages and torque command values

for the motor parameters presented earlier. Polynomials of 3rd and 5th order are used for the

main and coefficient equations, respectively. It can be seen that in the worst case the mutual flux

linkages error goes up to ten percent.


p.u.,T*en

p.u.,*mnλ

p.u.,mnλ

Fig. 6.7. Commanded and achieved mutual flux linkages vs. commanded torque for the

polynomial implementation approach.

The digitized values of the mutual flux linkages and torque commands can be used in the

implementation of equations (6.24) to (6.27). The higher than 1 powers of these variables, in

digital values, can be computed off line and stored in tables. These tables can later be utilized, at

the implementation level, to readily provide the higher powers of the variables instead of

performing time consuming multiplications. Assuming that *enT and *

mnλ are digitized using 8

bits, storing the higher powers of each variable requires 256x(2+3+...+ η ) bytes, where η is the

highest order of the respective variable. For the example provided in this section, η is equal to 3

for the main equations and 5 for the coefficient equations. Therefore, the memory required to

store the two tables is approximately 5 kilobytes. On the other hand, 16 multiplications and 16

additions are required to implement equations (6.24) to (6.27) for the case given in this example.

6.6 COMPARISON OF THE TWO APPROACHES

The lookup table approach provides a high level of accuracy in terms of enforcing the mutual

flux linkages and torque commands as compared to the two-dimensional polynomial approach.

Therefore, utilizing lookup tables results in meeting the specified voltage requirements of the

drive system, which is one of the key factors in the drive’s performance. The system is also

relatively fast since implementing it mainly involves reading data from a memory chip, and the

performance of minimal calculations. However, implementing this system requires a


considerable amount of digital memory capacity. For the example provided in this section 128

kilobytes of ROM is required to store the tables. The two-dimensional polynomial fit approach

provides an alternative to the former approach. It resolves the problem of large memory

requirements. Roughly 5 kilobytes of memory is required for the implementation scheme

presented in section 6.5. However, this system introduces a significant amount of error in terms

of enforcing the desired mutual flux linkages and torque. The error is more pronounced at higher

levels of mutual flux linkages and torque commands as seen from Fig. 6.7. The system is also

inherently more complicated to implement than the lookup table approach since it requires the

implementation of variable coefficient polynomials. Moreover, the computation of the command

values using two-dimensional polynomials slows down the system due to the numerous

multiplications that have to be performed. Other sources of error that are common to both

approaches are the digitization of the variables, limited bandwidth of the current controller, and

parameter variations.

6.7 CONCLUSIONS

Two practical implementation schemes are formulated, analyzed and verified by dynamic

simulation for the concurrent mutual flux linkages weakening and torque control of a PMSM.

The most straight forward scheme is to use lookup tables, which provides a high level of

accuracy. As a tradeoff, the method requires additional memory chips to be present on board to

store the tables. In another approach these lookup tables are approximated using two-

dimensional polynomials. The two-dimensional polynomial approximation method introduces

error in terms of achieving the desired mutual flux linkages and torque. The maximum

permissible torque command as a function of the mutual flux linkages command is studied, and

methods for taking this restriction into account are discussed. The maximum permissible torque

command decreases nonlinearly as the mutual flux linkages is weakened.


• Introduction of practical implementation schemes, based on lookup tables or equations,

for the concurrent mutual flux linkages weakening and torque control for PMSM

• Comparison of the two implementation schemes

• Analysis of the maximum possible torque as a function of mutual flux linkages

107

CHAPTER 7

Concurrent Mutual Flux Weakening and Torque

Control in the Presence of Parameter Variations

7.1 INTRODUCTION

It is shown in Chapter 6 that the q and d axis current commands can be implemented using two-

dimensional lookup tables or equations in order to enforce a desired mutual flux linkages and

torque command on a PMSM. The two-dimensional tables are prepared off-line using the

nominal parameters of the machine. However, some of the machine parameters deviate from

their nominal values during normal operation of the machine. These variations are due to

changes in temperature as well as magnetic saturation of the core, and result in mutual flux

linkages and torque enforcement error. The mutual flux linkages error may directly translate into

higher voltage requirements. The torque error results in a nonlinear torque response of the

system. The computation of the correct current commands involves the on-line solution of a

fourth order polynomial equation as discussed in Chapter 6. The coefficients of this polynomial

depend on machine parameters as well as the commanded values for mutual flux linkages and

torque. Theoretically, this fourth order polynomial can be solved on-line while its parameters are

calculated based on estimated values of machine parameters. However, this method is not

feasible for practical applications due to its intensive computational requirements. It is shown in

this chapter that lookup tables can be prepared offline to yield an offset for the current commands

in response to the variation of a desired parameter. Each of the final current commands in this

case is the sum of a nominal value and an offset. The nominal value is calculated based on the

nominal values of machine parameters. The offset is function of the estimated value of a specific

parameter as well as the mutual flux linkages and torque commands. The application of the

nominal current vector, along with the computed offset, results in greatly reduced errors in

mutual flux linkages and torque command enforcement. This is true for the full range of mutual

Chapter 7 Concurrent Flux Weakening and Torque Control With Parameter Variations 108

flux linkages control, and hence, for a wide range of operating speed. A large number of lookup

tables are required in order to accurately cover the full range of the varying parameter. An

alternative approach is to train neural networks to replace the tables. Simulation results show

that neural networks are well suited for this purpose. A comparison between the two approaches

is made in this chapter. The magnet flux linkages is the most critical parameter to be

compensated for in PMSM drives, and, therefore, it is used in most of this chapter's examples.

Other parameters can be treated in a similar way.


• Procedure for computing the current vector compensation offset using lookup tables

• Procedure for computing the current vector compensation offset using neural networks

• Comparison of the two approaches

The nature of the variations of different PMSM parameters is discussed in section 7.2. The

impact of parameter variations on mutual flux linkages and torque enforcement is studied in

section 7.3. The parameter compensation scheme, along with two implementation strategies, are

discussed in section 7.4. A comparison of the two implementation strategies is given in section

7.5. Dynamic simulation of a neural network based compensator is presented in section 7.6. The

procedures for estimating the q-axis inductance and rotor flux linkages are discussed in section

7.7. The conclusions are summarized in section 7.8.

7.2 THE VARIATION OF PARAMETERS IN PMSM

The stator resistance, rotor flux linkages and q-axis inductance are the only parameters of a

PMSM that vary significantly. The variation of the stator resistance does not have any material

impact on the operation of high performance PMSM drives. This is due to the fact that high

performance PMSM drives predominantly use current controllers that enforce a desired current

on the motor regardless of the value of stator resistance. The torque of a PMSM depends on the

q and d-axis inductances and currents as well as the rotor flux linkages. Also note that the

voltage drop across the phase resistance is generally small compared to the back emf in the flux


weakening range for most PMSM. The rotor flux linkages may decrease by as much as 20%

while the motor is running under normal operating conditions depending on the magnets used.

The residual rotor flux linkages declines as the temperature of the magnets rise. The q-axis

inductance decreases as the q-axis current increases. The q-axis path of some types of PMSM has

a relatively low reluctance which results in saturation of the path as q-axis current increases.

This is especially true for inset and interior PMSM which have low reluctance along their q-axis

paths. The saturation of the q-axis path results in declining q-axis inductance. The q-axis

inductance of such motors may vary by as much as 20% of its nominal value. The surface mount

PMSM is effectively immune to inductance variations due to the large effective air gap which

results in relatively high reluctance along both d and q axes.

7.3 THE IMPACT OF PARAMETER VARIATIONS

The mutual flux linkages and torque controller system is shown in Fig. 7.1.

Lookup Tablesor

Equations

PMSM

Drive SystemMotor

*eT

*mλ

*qi

*di

rθ

c,b,ai eT

mλ

Fig. 7.1. The mutual flux linkages and torque control system.

*mλ and *

eT are the mutual flux linkages and torque commands, respectively. *eT can be an

independent command value, or it may be the output of a controller acting on the difference

between the commanded and measured values of speed. *qi and *

di are the q and d-axis current

commands, respectively. The difference between stator and rotor current commands, which is

due to core losses, is ignored here; but this difference can be accounted for by utilizing the

conversion matrix given in (6.11). The current commands are calculated using lookup tables or

equations that are prepared offline using nominal parameters of the specific PMSM. The


measured values of the a, b and c phase currents, c,b,ai , are used by the PMSM drive system to

control current. The rotor angle, rθ , is used by the drive system to implement vector control.

mλ and eT are the actual mutual flux linkages and torque, respectively. The mutual flux

linkages and torque commands are achieved accurately as long as the actual and nominal

parameters of the machine match. The sources of error in this case are mainly digitization error

and current enforcement error due to the limited bandwidth of the current controller. In practice,

motor parameters vary depending on the operating conditions of the machine. These variations

cause a mismatch between the actual and commanded values of mutual flux linkages and torque.

The mutual flux linkages and torque errors, mλ∆ and eT∆ , respectively, are defined as,

m*mm λ−λ=λ∆ (7.1)

e*ee TTT −=∆ (7.2)

The variations of both afλ and qL have a direct impact on the system as described in Fig. 7.1,

and can cause significant errors in mutual flux linkages and torque. The equations relating the

mutual flux linkages and torque to the q and d-axis currents are given below,

2qq

2afdd

2m )iL()iL( +λ+=λ (7.3)

)ii)LL(i(P75.0T dqqdqafe −+λ= (7.4)

Figures 7.2 and 7.3 show the normalized values of mλ∆ and eT∆ as a function of the mutual

flux linkages and torque commands for a 20% drop in the rotor flux linkages for the system

shown in Fig. 7.1. The parameters of the PMSM used in this study are given in section 3 of

Chapter 6. It can be seen that the mutual flux linkages and torque errors are as much as 0.2 and

0.3 p.u., respectively. For the case where the actual and nominal parameters are identical, the

mutual flux linkages and torque errors are negligible due to the accuracy of the nominal tables.

The rotor flux linkages and q-axis compensation procedures are discussed in the next section.


p.u.,*mnλ

p.u.,T*en

p.u.,Ten∆

Fig. 7.2. Normalized torque error when p.u.8.0af =λ

p.u.,*mnλ

p.u.,T*en

p.u.,mnλ∆

Fig. 7.3. Mutual flux linkages error when p.u.8.0af =λ

7.4 PARAMETER COMPENSATION SCHEME

The direct calculation of the current commands from (7.3) and (7.4) involves the on-line

solution of a fourth order polynomial of di as explained in Chapter 6. The parameters of this

polynomial depend on the commanded values and machine parameters, and must also be

estimated on-line. The on-line solution of this polynomial results in a theoretically perfect

system assuming that accurate estimations of the respective parameters are utilized. However,

the method is not feasible for practical applications due to its intensive computational

requirements. A feasible alternative is to compute parameter-dependent offsets to be added to

the nominal current commands. The command offsets, which are computed on-line, compensate

for the effects of parameter variations. Fig. 7.4 gives a block diagram of this system.


Lookup Tables

or

Neural Networks

PMSM

Drive SystemMotor

OffsetGenerator

*eT

*mλ

*qi

*di

*qi

*di

++

++

Parameter

Estimator

eT

mλrθ

*qi∆*di∆

c,b,ai

Fig. 7.4. The mutual flux linkages and torque controller with compensation for a varyingparameter.

*di and

*qi are the nominal current commands. These current commands are calculated as

functions of the mutual flux linkages and torque commands for the nominal values of all

parameters as discussed in Chapter 6. The offset generator takes the mutual flux linkages and

torque commands, along with the estimated value of the varying parameter, as its inputs. Its

outputs are the offsets *qi∆ and *

di∆ which are to be added to the nominal current commands,*qi

and*di , respectively. A requirement of this system is that the varying parameter be estimated on-

line. Estimation procedures for rotor flux linkages and q-axis inductance are discussed later in

section 7.7. The offset generator can be implemented using lookup tables or neural networks as

discussed next in sections 7.4.1 and 7.4.2. Both implementation strategies are presented using

the rotor flux linkages as the varying parameter. The procedure for compensating for the q-axis

inductance is the same.

7.4.1 Lookup Table Approach

The lookup table approach for the implementation of *qi∆ as a function of *

eT , *mλ and the

estimated rotor flux linkages, afeλ , is discussed here. The same approach can be applied to *di∆ .

A lookup table providing*qi as a function of *

eT and *mλ can be prepared offline by solving

(7.3) and (7.4) for the nominal value of rotor flux linkages, afλ . Assuming that the actual and

nominal values of afλ are the same, then *qi and

*qi are the same and the required offset is zero.


Similar tables can be calculated for *qi as a function of *

eT , *mλ and every possible value of afλ .

In practice preparing lookup tables for 10 to 20 different values of afλ provides sufficient

control accuracy. The lookup table that provides *qi∆ as a function of *

eT and *mλ for a given

afλ can then be calculated by simply subtracting the table prepared for afλ from the table

prepared for the given afλ on an element by element basis. afeλ must be digitized in order to

implement this system. For each digitized value of afeλ the corresponding table that yields *qi∆

is selected. An identical approach is necessary for the implementation of *di∆ as a function of

*eT and *

mλ for the full range of afλ .

7.4.2 Artificial Neural Network Approach

Artificial neural networks (ANN) are very suitable for generating the type of compensation

offset discussed above. An implementation scheme for this task is presented and verified by

simulation in this section. The offset generator consists of two ANNs that are used to calculate

the *qi∆ and *

di∆ , independently. Fig. 7.5 shows the structure of a suitable ANN for generating

the required offsets. The inputs, *eT and *

mλ , are passed through a preprocessor before being

applied to the ANNs. The output of the preprocessor contains 5 quantities, *eT , *

mλ , *m

*eT λ , 2*

eT

and 2*mλ . Increasing the dimension of the input space to the ANN facilitates the training process

and results in a more accurate system. The five outputs of the preprocessor along with afeλ

constitute the inputs to the ANN. Each ANN has 9 hidden layer neurons and 1 output layer

neuron. The 2-layer feedforward ANN is trained using the back propagation algorithm. Several

two-dimensional tables, which provide the correct current vector offset commands for different

values of afeλ (in the range of .8 to 1 p.u.), are used to train each ANN.


Preprocessor

9 Neuron Hidden Layer

Single neuronoutput layer

Two-layerFeedforward

Network

or

afeλ *eT *

mλ

*di∆ *

qi∆

Fig. 7.5. The ANN based offset generator structure.

7.5 COMPARISON OF THE TWO APPROACHES

It is important to notice that the lookup table approach results in the most accurate

implementation scheme for the offset generators. The lookup tables represent accurate solutions

to the equations discussed earlier. On the other hand, neural networks are trained to mimic the

behavior of the tables, and therefore, cannot represent the tables accurately. However, as is

shown later in section 7.6, neural networks exhibit a sufficient capacity in replacing the lookup

tables in high performance motor drive applications. The lookup table approach provides an

inherently faster system as compared to the neural network approach. No multiplications are

required with lookup tables. However, 108 multiplications are required just to implement the

weights in two separate ANNs as shown in Fig. 7.5. Therefore, the timing requirements become

more stringent due to the increased demand on sequential software operations. Also, the

nonlinear activation functions must be implemented using additional lookup tables or using

dedicated integrated circuits. The main disadvantage of using lookup tables is the relatively

large amount of programmable memory required by the system. Two tables are required for each

digitized value of afeλ . Therefore, 128 Kb of memory is required to implement the offset tables

required for each digitized value of afeλ . This is based on 9 bit digitization for torque, and 7 bit


digitization for mutual flux linkages. Therefore, 4 bit digitization of afeλ necessitates 2

megabytes of programmable memory in addition to the 128 kb memory requirements of the

nominal command tables.

7.6 DYNAMIC SIMULATION

Fig. 7.6 shows simulation results for a speed control system that utilizes the neural network

approach discussed in section 7.4.2 for the concurrent control of mutual flux linkages and torque.

The torque command is the difference between the commanded and actual speeds multiplied by a

factor of 100. The mutual flux linkages is maintained at rated value until the phase voltage

requirement reaches 1 p.u. at which point flux weakening is initiated. The dotted lines in Fig. 7.6

represent command values and solid lines represent actual values. The speed commands are ± 3

p.u. The rotor flux linkages is assumed to have dropped 20 percent below its nominal value.

Mutual flux linkages weakening is initiated at a speed of 0.6 p.u. Neural networks are used to

provide both the nominal q and d-axis current commands and their respective offsets. The phase

current commands are enforced using a PWM system operating at 20 kHz with a gain of 300. It

can be seen that the mutual flux linkages and torque commands are enforced with high accuracy

for the full range of speed while the voltage requirements never exceed 1 p.u. This simulation

shows that artificial neural networks can be effectively utilized in implementing parameter

insensitive controllers for PMSM. In an independent set of simulations, the ANNs are trained to

compensate for the q-axis self inductance variations. Fig. 7.7 shows the respective simulation

results for the case where qL is 20% lower than its nominal value.


p.u.,*mnλ

p.u.,mnλ

p.u.,Vsn

p.u.,T*en

p.u.,T*en

p.u.,i,i qn*qn

p.u.,i,i dn*dn

p.u.,*rnω

p.u.,rnω

Time, s

Fig. 7.6. Commanded and actual values for speed, torque, mutual flux linkages, q and d-axiscurrents and phase voltage for a ± 3 p.u. speed command in the presence of 20 percent reduction

in rotor flux linkages.

.


p.u.,*mnλ

p.u.,mnλ

p.u.,T*en

p.u.,T*en

Time, s

Fig. 7.7. Commanded and actual values for mutual flux linkages and torque for a ± 3 p.u. speed

command in the presence of a 20 percent reduction in q-axis inductance.

7.7 ESTIMATION OF MACHINE PARAMETERS

The key parameters of a PMSM that need to be estimated are the q-axis inductance and the rotor

flux linkages. The q-axis inductance can be estimated as a function of the q-axis current. This

estimation can be performed offline and stored as a lookup table or a polynomial fit. A high

performance torque controller can then use the online estimation of the q-axis inductance in

calculating the appropriate offsets for the q and d-axis current commands. Reference [40]

demonstrates and verifies this procedure. Estimation of the rotor flux linkages is more involved

than the estimation of the q-axis inductance. One method for estimating the rotor flux linkages is

to first derive the relationship between rotor temperature and magnet flux linkages. An

estimation of rotor temperature can then be used to estimate the rotor flux linkages. However, an

accurate and reliable estimation of rotor temperature is required in this case. Measuring rotor

temperature is not a trivial matter as the rotor is not stationary. The rotor temperature can be

measured using infrared sensors, or using a thermal probe and a slip ring, or using a dynamic

thermal model of the machine [39]. All of these methods are expensive to implement, and are

not viable for low cost high volume motor drives. Reference [43] provides an innovative

procedure for estimating the rotor flux linkages for PMSM drives. It is shown that the change in


the rotor flux linkages is equal to the change in reactive power divided by speed and d-axis

current. The reactive power can be estimated using the d and q-axis currents and voltages. The

strength of this procedure is that the variations in stator resistance do not affect the estimation of

the rotor flux linkages. This procedure can only be applied to PMSM drives that utilize non-zero

d-axis current in their torque control strategy.

7.8 CONCLUSIONS

A procedure for the concurrent mutual flux weakening and torque control of PMSM in the

presence of parameter variations is introduced in this chapter. The procedure is based on an

offset generator for the q and d-axis current commands. The required offsets are calculated as a

function of mutual flux linkages and torque commands as well as an on-line estimation of a

varying PMSM parameter. Two implementation strategies based on lookup tables and neural

networks are presented and compared based on their respective implementation requirements. It

is shown that the lookup table approach results in a fast and accurate system, but requires a

significant amount of programmable memory. The neural network approach is shown to result in

a high level of accuracy without requiring much programmable memory. However, the neural

network approach results in an inherently slower system due to the large number of calculations

that need to be performed on-line. The results are verified by simulations of the systems.

In general, it can be concluded that artificial neural networks exhibit a sufficient capacity in

replacing the lookup tables in high performance motor drive applications.


• Procedure for computing the current vector compensation offset using lookup tables

• Procedure for computing the current vector compensation offset using neural networks

• Comparison of the two approaches

119

CHAPTER 8

Applications of a New Normalization Technique

8.1 INTRODUCTION

A new normalization technique is introduced in this chapter. This normalization technique is

particularly useful in the analysis and simulation of PMSM motor drives. The magnet flux

linkages and d-axis inductance are used as normalization bases. Therefore, these parameters, as

well as number of poles, are eliminated from all equations. This greatly simplifies the analysis

and simulation of PMSM motor drive performance. The standard normalization technique uses

the rated values of variables as its bases. Another innovative normalization technique, which

also results is simplified equations, has been introduced in [31]. A comparison of the new

normalization technique with the other two techniques shows the superiority of the former.

Several key derivations are presented in this chapter using the new normalization technique. It is

clearly seen that the elimination of two parameters from all equations results in analytical terms

that are easier to deduce, comprehend and simulate. However, the non-normalized form is the

preferred choice for demonstration purposes as it facilitates the understanding of the impact of

each parameter on system performance. Also, it is preferred to normalize the variables in all

plots using rated values of variables in order to make the plots easier to understand and study.

One of the applications of the new normalization technique is in the preparation of generalized

application characteristics. For example current and power factor vs. torque characteristics for

different control strategies depend only on the saliency ratio when these characteristics are

normalized using the new normalization technique.


• Introduction of a new normalization technique



Chapter 8 Applications of a New Normalization Technique 120

• Derivation of the maximum continuous current requirement of a PMSM drive operating with

constant power loss

• Derivation of the maximum possible torque for a given mutual flux linkages

• Introduction of generalized application characteristics for PMSM drives

Section 8.2 describes the details of the new normalization technique. This technique is

compared to two other normalization techniques in section 8.3. The maximum torque vs. speed

envelope for six different control strategies is derived in section 8.4. The maximum current

requirement of the drive is derived in section 8.5. The maximum possible torque for a given

mutual flux linkages is calculated in section 8.6. Section 8.7 introduces generalized application

characteristics for PMSM using the new normalization technique. The conclusions are presented

in section 8.8.

8.2 THE NEW NORMALIZATION TECHNIQUE

The base values for this normalization technique are given below:

db LL = (8.1)

afb λ=λ (8.2)

rrb ω=ω (8.3)

where bL , bλ and bω are the base values for the d-axis inductance, magnet flux linkages and

electrical speed, respectively, and rrω is the rated value for the rotor's electrical speed. The base

values chosen in (8.1) and (8.2) are chosen to eliminate dL and afλ from all equations. This

results in significant simplification of all equations as is demonstrated below. The rated values

for torque and current are not 1 p.u. in this normalization technique, but rated speed is always 1

p.u. The base values for key variables of a PMSM can be calculated from (8.1) to (8.3) as,

d

afb L

Iλ

= (8.4)

afrbV λω= (8.5)


bafeb IP75.0T λ= (8.6)

bbb VI5.1P = (8.7)

where bI , bV , ebT and bP are the base values for current, voltage, torque and power,

respectively. Normalized entities are hereafter denoted with the subscript n. The normalized q-

axis and d-axis inductances, as well as the normalized magnet flux linkages, are calculated

below:

ρ==d

qqn L

LL (8.8)

1L

LL

d

ddn == (8.9)

1afn =λ (8.10)

where ρ can be recognized as the saliency ratio of a PMSM. The key electrical equations of a

PMSM are described below in normalized format:

ω+

ρω−

ω

=

0

RI

I

1R

R1

I

Icn

rn

dn

qn

cn

rn

cn

rn

dsn

qsn(8.11)

+ω+

+ρω−

+ω=

0

)R

R1(

I

I

R)R

R1(

)R

R1(R

V

Vcn

snrn

dn

qn

sncn

snrn

cn

snrnsn

dsn

qsn(8.12)

The normalized torque, enT , as a function of qnI and dnI is given below,

qndnqnen II)1(IT ρ−+= (8.13)

The net power loss equation for a PMSM can be described as,


])I1()I[(R

1)II(RP 2

dn2

qn2rn

cn

2dsn

2qsnsnlmn ++ρω++= (8.14)

8.3 COMPARISON TO OTHER NORMALIZATION TECHNIQUES

The standard normalization technique is to divide every variable by its rated value. This

technique is widely used in many engineering fields, and has the advantage of making the rated

value of every variable appear as 1 p.u. The standard normalization technique has special appeal

in plotting of variables as it simplifies the understanding of plots, and helps generalize the

conclusions made from a system analysis or a plot. None of the motor drive parameters, other

than the number of poles, P, are eliminated using this technique, and therefore, it does not have

any special advantage when applied to PMSM drives. Note that all of the three normalization

techniques discussed in this section eliminate the number of poles from all equations. Another

innovative normalization technique is introduced in [31]. This technique is specifically geared

towards simplifying PMSM analysis. The base values for torque and current in this

normalization technique are:

dq

afb LL

I−

λ= (8.15)

bafeb IP75.0T λ= (8.16)

The base values for inductance and flux, bL and bλ , respectively, can be derived as:

)LL(P75.0L dqb −= (8.17)

afb P75.0 λ=λ (8.18)

Therefore, the torque equation, in this case, is reduced to:

dnqnqnen IIIT −= (8.19)

This normalization technique has the advantage of eliminating all parameters from the torque vs.

current equation as is seen in (8.19). However, only the rotor flux linkages and number of poles


are eliminated from all other equations. This technique fails for PMSM with saliency ratio of 1,

i.e. qd LL = , as the base value for current is infinity for such machines. This means that the

performance of the surface mount PMSM cannot be studied using this normalization technique.

The surface mount PMSM is used in the majority of high performance applications that utilize

PMSM. However, the new normalization technique presented in this chapter applies to all types

of PMSM, and it results in simplified derivations and simulations. One drawback of the new

normalization technique is that the rated values of variables are not 1 p.u. as is the case with the

standard normalization technique. This is further clarified in section 8.7.

8.4 DERIVATION OF MAXIMUM TORQUE VS. SPEED ENVELOPES

The maximum torque as a function of speed for a desired maximum power loss for each of the

ME, MTPC, ZDAC, CMFL, UPF and CBE control strategies is derived in this section. All

derivations are based on assumption (3.7) which simplifies the power loss equation as given

below:

])I1()I[(R

1)II(RP 2

dn2

qn2rn

cn

2dn

2qnsnlmn ++ρω++= (8.20)

The maximum torque vs. speed envelopes shown in simulation results given in Chapter 5 are all

prepared based on the analysis described in this section. It is shown that the normalization

technique used here significantly simplifies the derivations and simulations that would have

otherwise involved a much higher level of complexity due to the addition of two more

parameters. The normalized q and d axis currents that yield maximum possible torque at a given

speed and power loss are referred to here as qmnI and dmnI , respectively.

8.4.1 The Maximum Efficiency Control Strategy

The problem here is to find the maximum possible torque for a given lmP at a given speed. The

procedure is described below. It can be concluded from (8.13) that,


dn

enqn I)1(1

TI

ρ−+= (8.21)

By inserting (8.21) into (8.20), the equation describing torque as a function of dnI is derived as,

)EI.DI.CI.BI.A(Q

1T dn

2dn

3dn

4dn

2en ++++−= (8.22)

where,

cn

22rn

sn RRQ

ρω+= ,

cn

2rn

2

sn R

)1(RA

ωρ−+=

)R

)2(R)(1(2B

cn

2rn

snωρ−

+ρ−= ,

))1(4)1(1(R

P)1(RC 2

cn

2rn

lpn2

sn ρ−+ρ−+ω

+ρ−−=

)2(R

2)1(P2D

cn

2rn

lpn ρ−ω

+ρ−−=

lpncn

2rn P

RE −

ω=

The derivative of 2enT with respect to dnI in (8.22) is calculated and equaled to zero in order to

find the dnI that results in maximum torque. Taking the described derivative with respect to dnI

and equating it to zero results in the following polynomial equation,

0DI.C2I.B3I.A4 dn2dn

3dn =+++ (8.23)

The negative real root of equation (8.23), with smallest magnitude, is dmnI . By inserting dmnI

into (8.22) the maximum possible torque for a given speed and lmnP can be found.


8.4.2 The Maximum Torque per Unit Current Control Strategy

It can be concluded from (5.12) that the following relationship is valid for the MTPC control

strategy:

)1

1I(II dndn

2qn ρ−

+= (8.24)

By substituting (8.24) into (8.20) a second order polynomial equation of dnI that yields the d-

axis current magnitude for a given lmP can be found. This polynomial equation is given below:

0CI.BI.A dmn2dmn =++ (8.25)

where A, B and C are defined below:

cn

22rn

sn R)1(2

2RAρ+ω+=

2rn

2

cn

sn )1

2(R

11

RB ω

ρ−ρ++

ρ−=

lmncn

2rn P

RC −ω=

The solution to (8.25) can be inserted into (8.24) to calculate qmnI . The maximum possible

torque at a given speed can then be calculated by inserting dmnI and qmnI into the torque

equation (8.13).

8.4.3 The Zero D-Axis Current Control Strategy

In this case dnI is always zero. Therefore, the q-axis current at maximum power loss can be

calculated from (8.20) as:

22rncnsn

2rncnlmn

qmnRR

RPI

ρω+ω−= (8.26)


The maximum torque at a given speed and lmnP can then be found by inserting qnmI into the

torque equation (8.13).

8.4.4 The Unity Power Factor Control Strategy

In can be concluded from (5.15) that the following equation is true in the case of the UPF

control strategy:

ρ+−= )1I(I

I dndn2qn (8.27)


axis current magnitude for a given lmP for the UPF control strategy can be found. This

polynomial equation is given below:


where A, B and C are defined as:

)1(R

RRA

cn

2rnsn

sn ρ−ω+ρ

−=

cn

2rn

R)2(

Bωρ−=

lmncn

2rn P

RC −ω=


torque at a given speed can then be calculated by inserting dmnI and qmnI into the torque

equation (8.13).

8.4.5 The Constant Mutual Flux Linkages Control Strategy

In can be concluded from (5.17) that the following equation is true in the case of the CMFL

control strategy:


2

2dn2

qn)I1(1

Iρ+−= (8.29)


axis current magnitude for a given lmP for the CMFL control strategy can be found. This

polynomial equation is given below:


where A, B and C are defined below:

)1(RA 2sn −ρ=

snR2B −=

)PR

(C lmncn

2rn2 −ωρ=


torque at a given speed can be calculated by inserting dmnI and qmnI into the torque equation

(8.13).

8.4.6 The Constant Back EMF Control Strategy

The following relationship holds in the flux weakening region,

))I()I1((E 2qn

2dn

2rn

2mn ρ++ω= (8.31)

where mnE is the normalized desired back emf. On the other hand (5.26) can be cast in

normalized format as given below,

cnsn

2mn

sn

lmn2dsn

2qsn RR

ER

PII −=+ (8.32)

Solving (8.31) and (8.32) results in the following polynomial equation that yields the d-axis

current at maximum voltage for a given maximum power loss,


0CI.BI.A dpn2dpn =++ (8.33)

where,

21A ρ−=

2B =

1E

RRE

R

PC

2rn

2mn

cnsn

2mn2

sn

lmn2 +ω

−ρ−ρ=

The negative zero of this polynomial is dmnI . qmnI can be derived by inserting dmnI into (8.32).

The maximum possible torque at a given speed can then be found by inserting dmnI and qmnI

into the torque equation (8.13). The fundamental value of the maximum possible torque under

the SSV control strategy can be found by replacing mnE in the above derivations with the

normalized fundamental component of phase voltage.

8.5 DERIVATION OF MAXIMUM CURRENT REQUIREMENT WITH CPL

In this section the maximum continuous current requirement of a drive operating with constant

power loss is derived as a function of rated current. At rated speed the normalized value of speed

is 1 p.u. The d-axis current is zero by definition of rated torque for PMSM. Therefore, the total

power loss at rated speed and torque can be described as,

)I1(R

1IRP 2

srn2

cn

2srnsnlrn ρ++= (8.34)

where the subscript r denotes that the respective variable is at rated value. It is assumed here that

the maximum possible power loss at any speed is the rated power loss, lrnP . It is seen from Fig.

5.2 that the maximum current requirement happens at zero speed for a motor drive operating

with constant power loss. Core losses are zero at zero speed. Therefore, at zero speed,

2smnsnlrn IRP = (8.35)

where smnI is the maximum possible current. It can be concluded from (8.34) and (8.35) that,


)I1(R

1IRIR 2

srn2

cn

2srnsn

2smnsn ρ++= (8.36)

The maximum current can be calculated from (8.36) as:

5.0

cnsn

2srn

cnsn

2

smn )RR1

I)RR

1((I +ρ+= (8.37)

The ratio of maximum current to rated current can be written as given below using (8.37),

5.0

srn

smn )1(I

I α+= (8.38)

where,

cnsn2srncnsn

2

RRI

1

RR+ρ=α (8.39)

The ratio of maximum current to rated current for the prototype PMSM drive described in

Appendix I is 1.35 as calculated from equation (8.38). Therefore, the maximum possible current

for the prototype machine is 35% above rated torque for operation at a load duty cycle of 1 at

zero speed. This maximum current requirement remains valid at very low speeds as well. This

is due to the fact that maximum current vs. speed profile of a machine does not drop off

significantly at very low speeds. This can be seen from Fig. 3.2.

8.6 MAXIMUM POSSIBLE TORQUE AS A FUNCTION OF FLUX LINKAGES

The mutual flux linkages control strategy, described in section 2 of Chapter 6, imposes a

maximum permissible torque emnT on the system. This maximum torque is dependent on the

mutual flux linkages, mnλ , and decreases nonlinearly as the mutual flux linkages decreases. In

this section emnT is computed as a function of mnλ and ρ . The following three characteristics

of the system have been implicitly utilized in the derivations,


1>ρ (8.40)

0Idn < (8.41)

10 mn ≤λ≤ ,p.u. (8.42)

The basic equations for computing the current pair I,I qndn from ,T mnen λ are,

dnqnqnen II)1(IT ρ−+= (8.43)

2qn

2dn

2mn )I()1I( ρ++=λ (8.44)

Equation (8.43) yields,

1I)1(

TI

dn

enqn

+ρ−= (8.45)

Inserting (8.45) into (8.44), and rearranging the result, yields,

2dn

2dn

2mn

2en ]1I)1][()I1([)T( +ρ−+−λ=ρ (8.46)

The following variable changes simplify the derivations,

xI1 dn =+ (8.47)

yTen =ρ (8.48)

Substituting (8.47) and (8.48) into (8.46) yields,

222mn

2 ]1)1x)(1].[(x[y +−ρ−−λ= (8.49)

The final form of (8.49) can be expressed as,

222mn

2 ]1x].[x[y α++α−−λ= (8.50)

where α is given by,

1−ρ=α (8.51)


The mutual flux linkages and α can be dealt with as constants, and x can be considered the

independent variable of equation (8.50). Differentiating 2y with respect to x, and equating it to

zero, yields the px for which 2y is at an extremum. The extremum of enT can then be

computed from that of 2y . The differential of equation (8.50) has two zeros of which only one

yields a positive value for 2y . Solution of the problem yields the following equation from

which px can be computed.

1x0xx2 p2mnp

2p ≤=λ+β+− (8.52)

where,

1−ρρ=β (8.53)

Finally, the following equation provides the maximum permissible torque, emnT , as a function of

the mutual flux linkages, motor parameters and px .

]1x

.[]x[T p2

12p

2mnemn +

β−−λ= (8.54)

In brief, equation (8.52) provides px , from which equation (8.54) yields the maximum torque for

a given mutual flux linkages. Figure 6.1 is prepared using the procedure described above, but

the variables are plotted using the rated values of each variable as base value.

8.7 GENERALIZED PERFORMANCE CHARACTERISTICS

One of the applications of the new normalization techniques is in generating performance

characteristics that are applicable to a wide range of PMSM drives. For example the current vs.

torque relationship for the maximum torque per unit current control strategy only depends on the

saliency ratio, ρ , when normalized using the new normalization technique. This is evident from

equations (8.13) and (8.24). Therefore, this relationship is applicable to all PMSM with a given

ρ . Fig. 8.1 shows the normalized current, air gap flux linkages, power factor and d-axis current

vs. torque for the maximum torque per unit current control strategy for saliency ratios of 1, 2 and

3. This range of saliency ratio covers the full range of commercial PMSM.


Fig. 8.1. Generalized application characteristics for PMSM with 3,2,1=ρ for the maximum

torque per unit current control strategy.

p.u.,Isn

p.u.,mnλ

.f.p

p.u.,Idn

p.u.,Ten

123

123

321

ρ

p.u.,Isn

p.u.,mnλ

.f.p

p.u.,Idn

p.u.,Ten

123

123

321

ρ


Note that 1 p.u. of a variable shown in Fig. 8.1 does not correspond to the rated value of the

respective variable. For example 1 p.u. of current and torque as shown in Fig. 8.1 correspond to

17.6 A and 6.5 N.m., respectively, if applied to the prototype motor drive used in this

dissertation. The equivalence of 1 p.u. for each variable can be derived using the base values

described in equations (8.15) to (8.18). The rated current and torque for the prototype motor

drive used in this dissertation are 6.6 A and 2.4 N.m., respectively. Therefore, the rated current

and torque correspond to 0.375 and 0.37 p.u., respectively, in Fig. 8.1.

One of the distinct conclusions that can be made from Fig. 8.1 is that the power factor for

PMSM with the maximum torque per unit current control strategy is not affected very much by

the saliency ratio. The power factor shown in Fig. 8.1 is calculated assuming that the difference

between the back emf and the phase voltage is negligible. It is also seen that the current required

at any given torque decreases as the saliency ratio increases, and the required magnitude of d-

axis current increases at the same time. Fig. 8.2 shows the same performance characteristics as

shown in Fig. 8.1 but with the zero d-axis current control strategy. It is seen here that the power

factor drops significantly as the saliency ratio increases for the zero d-axis current control

strategy. The current vs. torque characteristic is not affected by saliency ratio in this case.


Fig. 8.2. Generalized application characteristics for PMSM with 3,2,1=ρ for the

zero d-axis current control strategy.

p.u.,Isn

p.u.,mnλ

.f.p

p.u.,Idn

p.u.,Ten

123

321

ρ

p.u.,Isn

p.u.,mnλ

.f.p

p.u.,Idn

p.u.,Ten

123

321

ρ


8.8 CONCLUSIONS

A new normalization technique is introduced in this chapter. The magnet flux linkages and d-

axis inductance are chosen as base values so that these two parameters are eliminated from all

equations. This normalization technique simplifies the analysis and simulation of a PMSM

motor drive. Comparison to other normalization techniques shows the superiority of the new

technique. Analytical terms for the maximum torque vs. speed envelopes for the ME, MTPC,

ZDAC, UPF, CMFL and CBE control strategies are derived in this chapter using the new

normalization technique. The maximum continuous current requirement of a motor drive

operating with constant power loss is derived as a function of the motor’s rated current. The

maximum possible torque as a function of desired mutual flux linkages is also derived. The new

normalization technique can be used to generate generalized application characteristics for

PMSM drives. This is demonstrated for the maximum torque per unit current and zero d-axis

current control strategies. It is clear through all these derivations that the new normalization

technique simplifies and facilitates the analysis and simulation of high performance PMSM

drives.


• Introduction of a new normalization technique



• Derivation of the maximum current requirement of a PMSM drive with constant power loss

• Derivation of the maximum possible torque for a given mutual flux linkages

• Introduction of generalized application characteristics for PMSM drives

136

CHAPTER 9

Conclusions and Recommendations for Future Work

9.1 CONCLUSIONS

The main achievements of this dissertation are summarized below:

• Analysis of the operational boundary of variable speed PMSM drives

• Introduction of a control strategy that automatically limits the operational boundary of a

motor drive

• Analysis and comparison of different strategies for high performance control of PMSM

• Introduction of implementation techniques for concurrent mutual flux linkages and torque

control in the flux weakening range of operation for PMSM

• Introduction of implementation techniques for concurrent mutual flux linkages and torque

control in the flux weakening range of operation for PMSM in the presence of parameter

variations

• Introduction of a normalization technique that simplifies the analysis and simulation of

PMSM drive performance.

The traditional method of defining and implementing the operational boundary of a motor drive

is to limit torque and power to rated values. It is shown here that the traditional method results in

under-utilization of the machine and puts the system at risk of excessive power losses. The

constant power loss concept is introduced and analyzed here as the correct basis for defining and

studying the operational boundary of any motor drive. The implementation strategy for the

proposed scheme is developed. All major control strategies for linear torque control for PMSM

are analyzed and compared based on the constant power loss concept. The ME, MTPC, ZDAC,

UPF and CMFL control strategies are the main possible choices for the lower than base speed

operating range. The CBE and SSV control strategies are the main possible choices for the

higher than base speed operating range. Each set of control strategies results in a unique

Chapter 9 Conclusions and Recommendations for Future Work 137

operational boundary and performance. Therefore, the comparison provided in this dissertation

allows for choosing the optimal control strategy for a particular application.

The implementation of the constant power loss controller is achieved using a feedback loop

around a linear torque controller for PMSM. The area of implementation strategies for linear

torque control in the lower than base speed operating range has been adequately covered in the

literature. The area of torque control in the flux weakening range is revisited here. Practical

implementation strategies for the concurrent flux weakening and torque control for PMSM are

introduced in this dissertation. Also, implementation strategies for the concurrent flux

weakening and torque control in the presence of parameter variation are presented.

A new normalization technique is introduced that simplifies the analysis of PMSM motor

drives. The new normalization technique is shown to simplify the performance analysis for

PMSM drives. All derivations and simulations presented in this dissertation are performed using

the new normalization technique.

The contributions of this dissertation are summarized below:

• A constant power loss (CPL) based control strategy to obtain the maximum torque vs. speed

envelope

• An implementation scheme for the proposed constant power loss controller

• Calculation of the appropriate power loss command and maximum torque for the CPL

controller applied to applications with cyclic loads

• Comparison of maximum possible torque as a function of maximum possible power loss for

different applications with cyclic loads



• Comparison of control strategies along the constant power loss operational envelope

• Comparison of control strategies inside the constant power loss operational envelope

• Derivation of maximum speed, current and torque as a function of maximum power loss and

machine parameters

Chapter 9 Conclusions and Recommendations for Future Work 138

• Introduction of two practical implementation schemes, based on lookup tables or equations,

for the concurrent mutual flux linkages weakening and torque control for PMSM

• Analysis of the maximum possible torque as a function of the desired mutual flux linkages

• Procedure for computing the current vector compensation offset using lookup tables or neural

networks for high performance control of PMSM in the presence of parameter variations

• Introduction of a new normalization technique that simplifies the analysis and simulation

PMSM drive performance



• Introduction of generalized application characteristics for PMSM drives using the new

normalization technique


9.2 RECOMMENDATIONS FOR FUTURE WORK

The concepts presented in this dissertation are all demonstrated using PMSM drives. However,

all of these concepts equally apply to all motor drives. The next major step is to apply the

constant power loss controller to high performance PM brushless dc, induction, switched

reluctance and synchronous reluctance drives. The performance of different control strategies

for these motor drives need to be analyzed and compared based on the constant power loss

concept. Also, the implementation schemes presented here for concurrent flux weakening and

torque control for PMSM should be applied to the other types of drives mentioned above.

139

APPENDIX I

Prototype PMSM Drive

A high performance prototype PMSM drive is utilized in experimentally verifying the key

results of this dissertation. The prototype is assembled on bread-board, and is developed entirely

in hardware with no software programmable components. The motor drive block diagram is

shown in Fig. 3.3. A 16 bit resolver to digital converter is used to generate rotor position

feedback. Only 8 bits of this feedback are utilized in practice. The speed feedback is derived by

filtering the least significant bit of the resolver to digital output. The torque command is

digitized using 8 bits, and the speed feedback is digitized using 5 bits. The digitized torque and

speed are the inputs to two 8 kb EEPROMs. Each of these EEPROMs outputs an 8 bit command

for current magnitude and angle with respect to rotor, respectively. A desired control strategy

must be written into these EEPROMs. A vector rotator uses the position feedback and the digital

values for current magnitude and angle commands to generate phase a, b and c current

commands. Phase a, b and c current commands are then enforced on the machine using three

independent 10 kHz PWM current controllers. Current feedback is provided by two LEM

modules. The power stage uses six IGBTs (600V, 50A), and is capable of handling 10 Amperes

at 250 Volts.

The interior PMSM parameters are:

qL = 12.5 mH, dL = 5.7 mH, afλ = 123 mWeber-turns, P= 4, sR = 1.2 Ω , cR = 416 Ohms,

dcV = 118 Volts (Bus voltage).

The rated values of the system are given below:

speed = 3500 RPM, current = 6.6 A, torque = 2.4 N.m., power = 890 W, power loss = 121 W,

core losses at rated operating point = 43 W, copper losses at rated operating point = 78 W.

Combined motor and load inertia=0.0019 Kg. 2m , friction coefficient=2.7e-4 N.m./Rad/s.

140

APPENDIX II

Measurement of PMSM Parameters

The parameters of the PMSM described in Appendix I are measured as described below:

sR : The phase resistance is measured by dividing the line to line resistance by two. The line to

line resistance is measured by applying a known current and measuring the voltage drop across

the line to line terminal, or using an Ohm meter.

afλλλλ : The machine is back driven at a known speed. The peak of the line to line voltage divided

by square root of 3 is the phase back emf in this case. Rotor flux linkages can then be calculated

using the following equation,

ω=λ E

af (1)

cR : First the PMSM, with leads open, is back driven at a relatively low speed using a dc motor.

Any speed in the range of 100 to 300 RPM is fine. A relatively low speed is chosen so that only

a negligible magnitude of core losses is present. Then the power input to the dc motor is

measured. This input power is almost entirely indicative of the sum of copper and brush losses

on the dc motor and net friction losses. The friction losses can then be calculated by subtracting

dc motor copper and brush losses from the net input power to the dc motor. Note that the dc

motor current and phase resistance are known. Brush losses are negligible in this case. The

friction torque of the combined dc and PMSM setup can be calculated by dividing the estimated

friction losses by speed. Then the PMSM is back driven at 3500 RPM (rated speed) using the

same dc motor. The power input to the dc motor at 3500 RPM is the sum of copper losses of the

dc motor, friction losses, core losses of PMSM and windage losses. The friction losses can be

estimated by multiplying speed with the friction torque calculated before. The windage losses of

the dc motor is assumed to be negligible. The windage losses of the PMSM is estimated using

Appendix II Measurements of PMSM Parameters 141

the viscous damping coefficient (N.m./kRPM) given in manufacturer’s data sheets. The input

power to the dc motor net of dc motor copper losses, friction losses and windage losses of

PMSM is core losses of the PMSM. The core loss resistance, as defined in Fig. 3.1, can then be

calculated using the following equation,

l

22af

c P

5.1R

ωλ= (2)

where lP is defined here as being the net core loss calculated earlier.

qL and dL : q and d-axis inductances are measured by shorting phases b and c of a PMSM, and

measuring inductance across phases a and b while the rotor is locked. With phases b and c

shorted, the d-axis current in rotor reference fame is eliminated assuming that the q-axis of the

rotor is aligned with phase a. Therefore, the d-axis portion of the PMSM electrical model does

not contain any useful information in this case as back emf is also zero. The electrical model of

the machine can then be described as given below,

aqrass iLiRv ω+= (3)

Note that the q-axis and phase a currents are the same in this case. The q-axis of the rotor can be

aligned with phase a by finding the rotor position that yields minimum peak to peak current in

response to a fixed sinusoidal voltage input applied to phases a and b. Once this minimum peak

to peak current in known qL can be identified using the following equation,

2s

2r

2q

2

pp

ppRL)

I

V( +ω= (4)

where ppV and ppI are the peak to peak values of the input voltage and phase a current,

respectively, and rω is the frequency of input voltage times π2 . Alternatively, qL , can be

estimated by measuring the electrical time constant of the system using the following equation,

s

qll R

L=τ (5)

Appendix II Measurements of PMSM Parameters 142

Note that the rotor must be locked while making all measurements described above. The same

procedure as described above yields dL if the rotor position is such that ppI is maximized in

response to a fixed sinusoidal voltage input.

The lowest possible frequency for the input voltage yields the most accurate estimation of

inductance since the impact of core losses are minimum at lower frequencies. A frequency of 60

Hz is chosen in the case of this dissertation. Therefore, the inductance measurements are subject

to some degree of error due to core losses. Core losses cause discrepancies between the

measured and actual parameters while using the electrical time constant technique for the

measurement of inductance as well. This is especially true when measuring dL where the rate

of rise of current in response to a step input voltage is maximum.

143

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150

Vita

Ramin Monajemy received his B.Sc. degree in electrical engineering in 1988 from Sharif

University of Technology, Tehran, Iran. He received his M.Sc. degree in electrical engineering

in 1993 from George Mason University, Fairfax, VA, where he studied the field of model-based

fault detection and isolation, and published two papers on that subject. His current research

areas involve the high performance control of motor drives.

control strategies and parameter compensation for permanent

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